m 


TEXT    BOOK 

ON 

RAILROAD  SURVEYING 


BY 

GEORGE  WELLINGTON  PICKELS,  C.E. 

INSTRUCTOR  IN  CIVIL  ENGINEERING 
UNIVERSITY  OF   ILLINOIS 

AND 

CARROLL  CARSON  WILEY,  C.E. 

ASSOCIATE  IN  CIVIL  ENGINEERING 
UNIVERSITY  OF  ILLINOIS 


FIRST  EDITION 
FIRST   THOUSAND 


NEW  YORK 
JOHN  WILEY  &  SONS,  INC. 

LONDON  :     CHAPMAN  &  HALL,  LIMITED 
1914 


'•/ 

/r& 


Copyright,  1913,  by 
G.  W.  PICKELS  AND  C.  C.  WILEY 


PUBLISHERS  PRINTING  COMPANY 
207-217  West  Twenty-fifth  Street,  New  York 


PREFACE 

THIS  book  was  written  primarily  for  the  use  of  students  in 
Railroad  Surveying,  and  is  practically  a  summary  of  the  notes 
used  by  the  authors  in  their  classes  for  a  number  of  years. 

The  aim  throughout  has  been  to  present  the  essentials  of  the 
subject  in  simple,  concise  form  and  to  give  modern  practical 
methods  only.  To  this  end  detailed  mathematical  solutions 
are  omitted,  obsolete  methods  abandoned,  and  only  the  more 
common  problems  outlined;  consequently  the  subject-matter 
occupies  considerably  less  space  than  the  ordinary  hand-book, 
but  at  the  same  time  covers  all  phases  of  the  subject  thoroughly. 

From  the  standpoint  of  the  instructor,  some  of  the  essential 
characteristics  of  the  book  are: 

(1)  A  co-ordination  of  the  entire  subject,  indicating  the  general 
order  and  the  solution  of  the  various  steps  in  the  work. 

(2)  The  explanation  of  reasons  for  or  conditions  under  which 
the  various  problems  arise. 

(3)  The  assumption  of  a  knowledge  of  elementary  surveying 
and  plane  trigonometry  on  the  part  of  the  reader. 

(4)  The  consequent  omission  of  detailed  mathematics.     Out- 
lined solutions  only  are  given,  and  hence  the  student  must  follow 
the  solution  of  a  problem  through  step  by  step  instead  of  sub- 
stituting blindly  in  some  formula  which  has  the  appearance  of 
being    suitable.     The    student    is    therefore    more    thoroughly 
trained  in  the  plan  of  attack,  and  consequently  can  more  readily 
handle  the  multitude  of  problems  which  he  can  not  find  "all 
worked  out"  in  this  or  any  other  book. 

Although  designed  as  a  book  for  student  use,  it  is  believed  that 
it  will  appeal  to  the  practicing  engineer  as  well,  and  the  latter's 
attention  is  respectfully  called  to  the  following  features: 

(1)  The  simplified  curve  nomenclature  as  recommended  by  the 
American  Railway  Engineering  Association  has  been  adopted. 

(2)  The  " Spiral"  has  been  co-ordinated  with  the  circular  curve, 
where  it  properly  belongs  under  modern  railroad  practice.     It 
will  be  noted  that  the  spiral  here  given  is  of  a  general  form  and  is 

iii 

287914 


IV  PREFACE 

equally  applicable  with  chords  of  any  desired  length  as  a  so- 
called  " Six-chord"  or  " Ten-chord"  spiral. 

(3)  " Turnouts"  are  presented  in  practical  form  as  actually 
used  on  steam  roads  in  America,  i.e.,  with  straight  frogs  and 
switches. 

Acknowledgment  is  hereby  made  of  the  courtesy  of  Prof. 
J.  C.  Nagle  in  permitting  the  use  of  Tables  5,  6,  8,  and  9,  which 
are  taken  from  his  " Field  Manual  for  Railroad  Engineers";  to 
Professor  Carhart,  for  Table  7,  which  is  taken  from  his  "Field 
Book  for  Civil  Engineers";  and  to  Prof.  Walter  Loring  Webb, 
for  Tables  9  and  10  from  his  work  on  "Railroad  Construction," 
which  appear  as  Tables  10  and  11  in  this  volume. 

GEORGE  WELLINGTON  PICKELS, 
CARROLL  CARSON  WILEY. 

University  of  Illinois,  Urbana,  111.,  August,  1913. 


CONTENTS 

CHAPTER  I 
RAILROAD  SURVEYS 

PARAGRAPH  PAGE 

1.  Terminals    and    Intermediate    Points    Connected    by 

a  Railroad 1 

2.  The  Reconnoissance  Survey 1 

3.  Reconnoissance  in  Prairie  Country     ......  2 

4.  Reconnoissance  in  Rolling,  Hilly  Country    ....  2 

5.  Reconnoissance  in  Mountainous  Country     ....  2 

6.  Preliminary  Survey  in  Prairie  Country 3 

7.  Preliminary  Survey  in  Rolling,  Hilly  Country  ...  3 

8.  Preliminary  Survey  in  Mountainous  Country    ...  4 

9.  The  Location  Survey 4 

10.  Reference  Stakes 4 

11.  Slope  Stakes 5 

12.  Distribution  Stakes 6 

13.  Borrow-Pit  Stakes 6 

14.  The  Location  of  Culverts  and  Trestles 6 

15.  The  Time  to  Set  Stakes 6 

16.  Monthly  Estimate  Surveys 6 

17.  Finishing  Stakes 6 

18.  Center  Stakes 6 

19.  Grade  Stakes 6 

20.  Right  of  Way  Stakes 6 

21.  Special  Surveys 7 

22.  Maintenance  Surveys 7 

23.  Monumenting 7 

24.  Suryej^s  for  Additional  Tracks 7 

25.  The  Transit  Party 7 

26.  The  Level  Party 9 

27.  The  Topography  Party 9 

28.  The  Land-line  Party 9 

29.  Drafting 9 

v 


VI  CONTENTS 

PARAGRAPH  PAGE 

30.  The  Transit  and  Tape  Method  of  Making  Preliminary 

Surveys 10 

31.  The  Transit  and  Stadia  Method  of  Making  Preliminary 

Surveys 10 

32.  The    Plane    Table    Method   of   Making   Preliminary 

Surveys 10 

33.  Remarks 11 

34.  Bench  Marks 11 


CHAPTER  II 
MAPS 

35.  The  Reconnoissance  Map 12 

36.  The  Reconnoissance  Profile 12 

37.  The  First  Preliminary  Maps    .  12 

38.  The  Preliminary  Profiles 12 

39.  The  Preliminary  Map  (proper)      ....'...  12 

40.  The  Projected  Location 13 

41.  Location  Notes 13 

42.  The  Location  Map 13 

43.  The  Location  Profile 14 

44.  Additional  Profiles 14 

45.  Legal  Right  of  Way  Maps 15 

46.  Maintenance  of  Way  Right  of  Way  Maps    ....  15 

47.  Station  Maps 15 

48.  Progress  Profiles 15 

49.  Progress  Photographs 15 


CHAPTER  III 
DISTANCE,  CURVATURE,  AND  GRADES 

50.  Additional  Distance 16 

51.  Effect  of  Curvature 17 

52.  Choice  of  Curvature 17 

53.  Minor  Grades 18 

54.  Ruling  Grades 18 

55.  Pusher  Grades .  19 

56.  Choice  of  Grades 19 

57.  Vertical  Curves 21 


CONTENTS  Vll 

CHAPTER  IV 

CURVES 
Part  1.     Circular  Curves 

PARAGRAPH  PAGE 

58.  Functions  of  a  Simple  Curve 24 

59.  Relation  between  R  and  D 26 

60.  Field  Work   . 27 

61.  Fundamental  Principles 32 

62.  Offsets  from  Chords  Produced >33 

63.  Chord  Offset  Method .  34 

64.  Example 35 

65.  To  Find  New  Radius  for  a  Given  Change  in  E       .      .  36 

66.  To  Change  T.C.  to  Make  C.T.  Fall  on  ParaUel  Tangent  37 

67.  To  Change  R  and  T.C.  to  Make  C.T.  FaU  on  ParaUel 

Tangent,  on  same  Radial  Line,  Approximately     .  38 

68.  To  Change  R  to  Make  T.C.  Fall  on  Parallel  Tangent  38 

69.  To  Shift   C.T.   so   that   Forward   Tangent  will  Pass 

Through  Controlling  Point 39 

70.  Compound  Curves 41 

71.  Given   the   Central   Angles   and   Radii   to   Find   the 

Tangents 43 

72.  Given  both  Tangents  and  one  Radius  to  Find  the  Other 

Radius 43 

73.  Field  Work 45 

74.  To  Change  Location  of  C.C.  so  that  C.T.  will  FaU 

on  ParaUel  Tangent 45 

75.  To  Change  Second  Radius  and  the  Location  of  C.C. 

so  that  C.T.  wiU  FaU  on  Parallel  Tangent,  on  Same 

Radial  line,  Approximately 46 

76.  Reversed  Curves 49 

Part  2.     Spirals 

77.  Definition  of  Spiral '49 

78.  Functions  of  Spiral 50 

79.  Formulas 52 

80.  Length  of  Spiral 52 

81.  Field  Work 55 

82.  Deflection-Angle  Method 56 

83.  Offset  Method 60 

84.  To  Insert  Spiral  between  the  Branches  of  Compound 

Curve •..,-.- 62 

85.  Deflection  Angles  from  Point  on  Spiral 62 

86.  To  Spiral  Compound  Curve  at  Each  End  and  between 

Branches   ...  63 


Vlll  CONTENTS 

PARAGRAPH  PAGE 

87.  Application  of  Spiral  to  Existing  Simple  Curves    .      .  64 

88.  To  Spiral  Existing  Simple  Curve  by  Shifting  Track 

outward  near  Center 64 

89.  To    Spiral    Existing    Simple    Curve  by  Sharpening 

Curve  at  Ends 66 

90.  To  Spiral   Existing  Simple  Curve  by  Compounding 

Curve  near  Center 67 

91.  Application  of  Spiral  to  Existing  Compound  Curves  .  68 

CHAPTER  V 
EARTHWORK    • 

92.  Railroad  Cross-Sections 73 

93.  Width  of  Roadbed 73 

94.  Ditches 74 

95.  Side  Slopes 74 

96.  Borrow  Pits 74 

97.  Shrinkage 75 

98.  Growth '  .      .  75 

99.  Settlement 75 

100.  Slope  Stakes 76 

101.  Cross-Sectioning 77 

102.  Field  Work 77 

103.  Sections  at  Grade  Points 80' 

104.  Side-Hill  Sections 82 

105.  Uncompleted  Sections 82 

106.  End  Area  Method  of  Computing  Volumes       ...  83 

107.  Prismoidal  Method  of  Computing  Volumes      ...  84 

108.  Earthwork  on  Curves 85 

109.  Computation  of  Haul 88 

110.  Problem  in  the  Distribution  of  Earth 87 

111.  Problem,  to  Find  Limit  of  Profitable  Haul       ...  89 

112.  Effect  of  Trestle?,  etc.,  on  the  Distribution  of  Earth  90 

CHAPTER  VI 

TURNOUTS,  CONNECTIONS,  AND  CROSSINGS 

113.  Outline  of  Subjects  Treated 91 

114.  Definitions 91 

115.  Switches 92 

116.  Frogs 94 

117.  Frog  Angles 96 

118.  Frog  Numbers 96 


CONTENTS  IX 
PARAGRAPH                                                                                                                   PAGE 

119.  Location  of  Turnouts 96 

120.  Turnouts  from  Straight  Track 96 

121.  Turnouts  from  Curved  Track     ., 97 

122.  Double  Turnouts 98 

123.  Practical  Leads 98 

124.  Diverging  Tracks,  from  Straight  Tracks     ....  99 

125.  Diverging  Tracks,  from  Curved  Tracks      ....  100 

126.  Sidings  from  Straight  Tracks 104 

127.  Sidings  from  Curved  Tracks 106 

128.  Crossovers  between  Straight  Tracks 109 

129.  Crossovers  between  Curved  Tracks Ill 

130.  Connection    between    Intersecting    Straight    Tracks  111 

131.  Connection  between  One  Straight  and  One  Curved 

Intersecting  Track .      .113 

132.  Connection  between  Intersecting  Curved  Tracks        .  115 

133.  Crossings 116 

134.  Crossing  Data 117 

135.  Straight  Crossings 118 

136.  Single-Curve  Crossings .  119 

137.  Double-Curve  Crossings 120 

APPENDIX.    THEORY  OF  SPIRAL 

138.  Derivation  of  3  and  A 121 

139.  Derivation  of  the  Coordinates,  x  and  y  121 

140.  Derivation  of  a,  A,  b,  B ;     .      .  122 

141.  Derivation  of  0 123 

142.  The  Osculating  Circle 124 

143.  Superelevation 124 

TABLES 

1.  Spiral  Table 70 

2.  Turnout  Table 99 

3.  Siding  Table 105 

4.  Crossover  Table. 110 

5.  Table  of  Radii 128 

6.  Functions  of  a  1°  Curve 131 

7.  Logarithms  of  Numbers 158 

8.  Logarithmic  Sines  and  Cosines 176 

9.  Logarithmic  Tangents  and  Cotangents 191 

10.  Natural  Sines,  Cosines,  Tangents  and  Cotangents  .     .  206 

11.  Natural  Versed  Lines  and  External  Secants  .     .      .      .  229 

12.  Cubic  Yards  per  100  ft.  in  Terms  of  Center  Height      .  252 

13.  Cubic  Yards  per  100  ft.  in  Terms  of  Sectional  Area      .  258 

14.  Trigonometric  Formulas 262 


TEXT  BOOK  ON 
RAILROAD  SURVEYING 

CHAPTER  I 
RAILROAD  SURVEYS 

THE  practice  in  railroad  location  surveys  in  one  part  of 
the  United  States  is  entirely  different  from  that  in  another  part, 
so  that  rules  which  will  apply  in  one  locality  will  not  apply  at 
all  in  others.  All  country  can  be  broadly  classed  in  three 
divisions:  (1)  level  prairie  country  offering  no  obstacles  in  the 
way  of  hills,  valleys,  etc.,  and  allowing  the  locating  engineer 
much  latitude  in  the  placing  of  his  line,  and  consequently  re- 
ducing distance  and  curvature  to  a  minimum;  (2)  rolling,  hilly 
country  through  which  several  lines  are  possible,  none  of  them 
departing  to  any  great  extent  from  the  direct  line  between 
controlling  points;  and  (3)  mountainous  country,  which  taxes 
the  skill  and  ingenuity  of 'the  locating  engineer  to  the  utmost. 

Reconnoissance  Surveys 

1.  The  terminals  and  intermediate  points  connected  by  a 
railroad  are  determined  by  its  promoters  with  a  view  to  the 
amount  of  traffic  that  can  be  expected  from  them.     The  nature 
and  the  amount  of  the  traffic  and  the  direction  of  heaviest  haul 
determine   the   maximum   gradients   and   curvature   advisable. 
The  first  thing  to  be  determined  by  the  locating  engineer  is  the 
directions  of  the  lines  joining  controlling  points.     This  informa- 
tion can  usually  be  obtained  with  sufficient  accuracy  from  a 
map  of  the  territory  which  it  is  generally  possible  to  obtain. 
The  topographic  maps  published  by  the  government  are  the 
best. 

2.  A  reconnoissance  is  then  made  of  the  strip  of  country 
through  which  the  road  is  to  pass,  as  the  result  of  which  some  of 

1 


2  TEXT  BOOK   ON   RAILROAD   SURVEYING 

the  routes  are  eliminated  as  impractical  and  one  or  more  chosen 
for  a  more  detailed  survey. 

3.  If  the  country  is  of  the  first  class,  no  reconnoissance  is 
necessary  and  the  survey  is  in  the  nature  of  a  preliminary. 

4.  For  country  of  the  second  class,  the  reconnoissance  should 
be  made  across-country  on  horseback,  following  the  direction 
of  the  line  determined  from  the  map.     If  in  following  the  direct 
line  between  controlling  points  obstacles  are  met  with  which 
can  not  be  surmounted,  such  as  high  hills,  the  engineer  should 
explore  on  both  sides  of  the  obstacle  and  decide  if  possible  which 
route  offers  the  least  resistance.    Before  deflecting  from  a  straight 
line,  the  engineer  must  be  sure  that  his  reasons  for  so  doing  are 
justified    from    an    economic    standpoint.     In    country    of   the 
second  class  it  is  seldom  that  grades  cause  much  trouble,  and  the 
main  lookout  of  the  engineer  is  to  keep  the  amount  of  curvature 
as  low  as  possible.     The  result  of  the  reconnoissance  through 
country  of  the  second  class  is  that  one  or  more  routes  are  selected 
for  a  more  detailed  survey. 

5.  It  is  in  mountainous  country — third  class — that  all  the 
skill  of  the  engineer  is  brought  into  play.     The  drainage  of  the 
country  should  be  carefully  studied,  as  it  plays  a  very  important 
part  in  the  location  of  a  railroad.     If  the  controlling  points  are 
in  the  same  valley,  the  main  problem  is  solved;  and  the  con- 
forming of  the  alinement  to  the  topography  is  merely  a  matter  of 
detail.     But  when  the  controlling  points  are  in  different  valleys, 
the  ridges  between  them  have  to  be  crossed,  and  the  principal 
object  of  the  reconnoissance  is  to  discover  the  most  favorable 
crossing  places  in  the  valleys  and  on  the  ridges.     The  saddles 
in  the  ridges  and  the  most  favorable  river  crossings  become 
secondary  controlling  points.     The  location  of  all  such  points 
is  platted  on  the  map,  and  their  elevations  and  distances  apart 
are  recorded.     The  most  important  and  useful  "  instruments  " 
used  on  reconnoissance  are  the  judgment  and  experience  of  the 
locating  engineer,  as  upon  these  depend  the  amount  and  cost 
of  more  detailed  surveys  and  the  cost  of  construction  and  of 
operation. 


RAILROAD   SURVEYS  3 

Preliminary  Surveys 

6.  In  country  of  the  first  class,  a  preliminary  line  is  run  for 
direction.     The  magnetic  bearing  of  the  direct  line  between 
controlling  points  is  scaled  off  the  map,  and  a  line  having  this 
direction  is  initiated  from  the  first  controlling  point  and  pro- 
duced to  the  second  controlling  point.     Since  the  direction  of 
the  line  as  obtained  from  the  map  and  this  direction  as  laid  off 
in  the  field  are  subject  to  considerable  error,  the  first  line  run 
will  probably  pass  to  one  side  of  the  second  controlling  point. 
The  distance  by  which  the  transit  line  misses  its  mark  is  noted 
and  the  correction  that  must  be  applied  to  the  first  line  is  com- 
puted.    The  next  line  run  will  be  the  location. 

In  running  the  preliminary  line  stakes  should  be  placed  only  at 
transit  points.  A  straight  line  is  the  most  difficult  one  to  run, 
and  particular  care  must  be  taken  to  avoid  errors.  It  must  be 
remembered  that  the  preliminary  is  a  reference  line,  and  if  it 
is  not  straight  the  purpose  for  which  it  is  run  is  defeated.  The 
distances  between  hubs  can  be  determined  with  sufficient 
accuracy  by  means  of  the  stadia. 

7.  In  country  of  the  second  class,  preliminary  lines  are  run 
over  each  of  the  routes  chosen  by  the  locating  engineer  on  the 
reconnoissance   survey.     The   data   taken   are   such   that   the 
several  routes  can  be  compared  with  respect  to  distance,  grades, 
and  curvature.     Usually  lines  run  with  the  transit  and  stadia 
will  give  sufficient   data   as  regards   distance   and   curvature; 
and  the  elevations  of  enough  commanding  points  can  be  taken 
with  the  stadia  to  indicate  the  grades  that  will  be  required  by 
the  several  routes.     From  these  data  one  of  the  routes  will 
usually  appear  superior  to  the  others,  and  what  is  generally 
known  as  a  preliminary  survey  is  then  made  over  the  selected 
route.     A    preliminary    survey    as    generally    understood    is    a 
topographic  survey  of  a  narrow  strip  of  country  within  which  the 
road  must  pass.     The  purpose  of  this  survey  is  to  secure  data 
from  which  a  topographic  map  can  be  platted  upon  which  the 
paper  location  is  projected.     The  transit  and  stadia  line  already 
run  over  the  chosen  route  is  used  as  the  base  line  from  which 
the  topography  is  taken.     Although  this  line  is  usually  measured 
with  the  tape,  time  used  for  this  purpose  is  wasted,  as  it  is 
impossible  to  plat  the  traverse  distances  to  a  consistent  degree 


TEXT   BOOK   ON   RAILROAD   SURVEYING 

of  accuracy.  Spirit  levels  are  then  run  over  the  line  to  determine 
the  elevations  of  the  transit  points,  and  bench  marks  are  es- 
tablished at  half-mile  intervals. 

8.  In  country  of  the  third  class,  several  preliminary  lines  are 
required,  each  following  one  of  the  routes  chosen  by  the  locating 
engineer.     The  approximate  grades  of  the  lines  between  secondary 
controlling  points  are  obtained  from  the  data  taken  on  the  re- 
connoissance  survey,  and  the  preliminary  line  should  be  chosen 
so  that  the  grade  line  so  determined  will  conform  as  nearly  as 
possible  to  the  surface  of  the  ground.     If  this  is  done,  topography 
will  not  have  to  be  taken  as  far  on  each  side  of  the  line  as  would 
otherwise  be  necessary.     In  mountainous  country  topography 
can  be  taken  more  accurately  with  the  hand  level  than  with  the 
stadia,  and  line  stakes  are  placed  every  100  feet  and  their  eleva- 
tions determined  by  spirit  leveling  so  that  this  can  be  done. 
In  other  respects  the  methods  of  surveying  in  mountainous  coun- 
try are  the  same  as  those  used  in  country  of  the  second  class. 

Location  Survey 

9.  On  location  the  line  is  run  very  carefully,  stakes  are  driven 
every  100  feet,  the  plusses  and  angles  of  all  property  lines  are 
taken,  and  curves  are  run  in  where  indicated  on  the  map.     The 
distances  from  intersections  with  property  lines  to  the  nearest 
government  section  monument,  or  other  legal  monument,  are 
carefully  chained,  so  that  accurate  descriptions  of  the  right  of 
way  can  be  drawn  up.     All  buildings  near  the  line  which  will  be 
damaged  by  it  must  be  located  and  an  estimate  of  the  damage 
be  made.     A  level  party  follows  the  transit  party  taking  profile 
levels  and  establishing  bench  marks  unless  previously  established. 
From  these  data  alinement  maps,  right  of  way  maps,  and  pro- 
files are  made;  and  after  the  grade  line  is  established  on  the 
profile,  construction  can  begin. 

Construction  Surveys 

10.  Reference  stakes.     After  the  location  has  been  made  and 
accepted,  and  just  before  construction  begins,  the  beginning  and 
end  of  all  curves  and  intermediate  points  on  long  tangents  are 
"  tied-in  "  by  reference  stakes  so  that  after  the  construction 


RAILROAD   SURVEYS  5 

woik  is  completed  these  points  can  be  re-located  in  their  correct 
positions.  There  are  several  methods  of  referencing  a  point; 
the  one  shown  in  Fig.  1  is  very  satisfactory.  A  is  the  point 
to  be  referenced.  First  choose  permanent  points  at  C  and  E 
at  least  300  feet  from  A,  making  the  angle  CAE  as  nearly  a 
right  angle  as  possible.  A  distant  windmill  or  house-chimney 
makes  an  ideal  point.  If  these  are  not  to  be  had,  a  nail 


FIG.  i. 

driven  into  the  trunk  of  a  tree  about  five  feet  from  the  ground 
is  good.  If  hubs  are  used  they  should  be  made  as  permanent 
as  possible,  and  located  so  that  they  will  not  be  disturbed.  Af- 
ter C  and  E  have  been  determined,  set  a  hub  at  point  B  on  the 
line  AC,  and  another  one  at  D  on  the  line  AE.  B  and  D  should 
be  placed  far  enough  from  the  center  line  of  the  road  so  that 
they  will  not  be  disturbed  during  construction.  Point  A  is  re- 
located after  construction  by  the  intersection  of  the  lines  CB 
and  ED,  the  transit  being  set-up  at  B  and  D.  Care  and  judg- 
ment should  be  used  in  locating  reference  points,  as  they 
should  be  used  after  the  roadbed  has  settled  for  locating  per- 
manent monuments. 

11.  Slope  stakes  are  then  set  at  each  station  on  each  side  of 
the  center  line  at  the  points  where  the  side  slopes  of  the  cut  or 
fill  will  intersect  the  ground  surface.  These  stakes  are  for  the 
guidance  of  the  contractor,  and  have  marked  on  them  the 
vertical  distance  from  the  ground  at  the  stake  to  the  level  of  the 
roadbed.  (For  method  of  setting  slope  stakes,  see  paragraph  100.) 


6  TEXT   BOOK   ON   RAILROAD   SURVEYING 

12.  Distribution  stakes  are  set  to  show  the  contractor  the 
desired  movement  of  the  earth  from  the  cuts  into  the  fills.     (See 
paragraph  108.) 

13.  Borrow-pit  stakes  are  set  to  indicate  to  the  contractor  the 
limits  within  which  he  may  borrow  earth  for  making  the  fills. 

14.  The    location    of   culverts    and    trestles    must    also  be 
staked    out.     If    possible,     these    should    be    constructed  in 
advance  of   the  grading  so  that  openings  will  not  have  to  be 
left  in  the  fill  to  be  completed  later. 

15.  The  time  to  set  stakes  is  just  before  they  are  needed.     If 
they  are  set  too  long  a  time  ahead,  some  of  them  will  invariably 
be  knocked  out  and  will  require  re-setting. 

16.  Monthly  estimate  surveys.     During  the  last  few  days  of 
every  month  surveys  are  made  to  determine  the  amount  of  earth 
moved  by  the  contractor  during  the  month,  so  that  he  can  be 
paid. 

17.  Finishing  stakes.     From  the  slope  stakes  the  contractor 
can  construct  the  roadbed  to  within  a  few  inches  of  the  correct 
grade.     At  this  stage  of  the  work  it  is  customary  to  give  finishing 
stakes,  which  are  stakes  driven  to  grade  at  the  edges  of  the 
roadbed  at  each  station.     From  these  the  contractor  is  able  to 
finish  the  roadbed  to  the  correct  grade  and  width. 

18.  Center  stakes.     After  the  earthwork  is  completed,   the 
important  points  on  the  center  line  are  re-located  from  the 
reference  stakes,  and  center  stakes  (usually  untacked)  are  driven 
from  which  the  track  is  laid. 

19.  Grade  stakes.     After  the  track  is  laid,  grade  stakes  are 
driven  at  every  station  and  at  those  points  where  the  grade 
changes  with  their  tops  to  the  grade  of  the  final  top  or  base  or 
rail.     Grade  stakes  are  placed  on  the  inside  of  curves. 

20.  Right  of  way  stakes.     The  right  of  way  fences  are  usually 
built  as  soon  as  the  materials  for  construction  can  be  hauled 
over  the  line.     For  the  guidance  of  the  fence  foreman,  stakes  are 
placed  on  each  right  of  way  line  (1)  opposite  the  beginning  and 


RAILROAD    SURVEYS  7 

end  of  all  curves,  (2)  opposite  each  station  on  curves,  (3)  from 
300  to  500  feet  apart  on  tangents,  and  (4)  at  all  jogs  in  the  right 
of  way.  These  stakes  should  be  long  enough  to  be  seen  above 
the  weeds,  wheat,  oats,  etc.  Laths  are  excellent  for  this 
purpose. 

21.  Special  surveys.     After  the  track  is  laid,  special  structures, 
such  as  station  buildings,  water  tanks,  cattle  pens,  etc.,  will 
require  staking  out.     In  addition,  the  parts  of  the  right  of  way 
leased  to  coal,  lumber,  and  grain  companies  must  be  staked. 

Maintenance  Surveys 

22.  Due  to  the  fact  that  it  takes  two  or  three  years  for  the 
roadbed  to  settle  and  for  the  track  to  become  thoroughly  em- 
bedded in  the  ballast,  it  will  be  necessary  to  re-set  center  and  grade 
stakes  frequently  during  this  period.     All  center  stakes  which 
are  set  after  the  track  is  laid  should  be  tacked. 

23.  Monumenting.     After  the  track  and  roadbed  have  settled 
thoroughly,   permanent   monuments  should   be   placed   at   the 
beginning  and  end  of  all  spirals  and  circular  curves,  between 
the  branches  of  compound  curves,  and  at  intermediate  points  on 
long  tangents. 

24.  Additional  tracks,  such  as  side  tracks,  business  tracks, 
•branch-line  tracks,  yard  tracks  of  various  kinds,   cross-overs, 

etc.,  are  required  to  be  staked  out  from  time  to  time.  There  is 
no  end  to  surveys  of  this  kind,  and  all  the  large  railroads  employ 
maintenance  parties  who  do  nothing  else. 

Organization  of  Parties 

The  field  corps  is  usually  divided  into  (1)  a  transit  party, 
(2)  a  level  party,  (3)  a  topography  party,  and  sometimes  (4)  a 
land-line  party. 

25.  TRANSIT  PARTY.— The  members  of  the  transit  party 
and  their  duties  are  as  follows: 

The  Locating  Engineer  is  the  chief  of  the  entire  surveying 
corps,  and  receives  his  instructions  from  and  reports  to  the  Chief 


8  TEXT   BOOK   ON   RAILROAD   SURVEYING 

Engineer  of  the  railroad  company.  His  duties  are:  to  direct 
all  the  surveys  from  the  reconnoissance  to  the  location,  to  pro- 
vide accommodations  for  his  party,  to  pay  all  general  expenses, 
and  in  case  a  camp  is  necessary  to  purchase  all  supplies  and  to 
manage  the  camp. 

The  Transitman  is  next  in  rank  to  the  locating  engineer,  and  in 
his  absence  is  in  charge  of  the  party.  His  duties  are:  to  do  the 
transit  work,  which  consists  of  lining  in  the  chainmen,  measuring 
the  angles  between  successive  tangents,  noting  the  bearings  of 
the  tangents,  measuring  the  angles  which  the  line  makes  with 
all  railways,  highways,  streams,  and  property  lines,  and  record- 
ing the  plusses  at  which  they  cross  the  line;  and  to  keep  the  notes 
of  the  transit  party.  On  construction  the  transitman  usually 
becomes  Resident  Engineer  and  has  charge  of  from  8  to  15 
miles  of  construction. 

The  Head  Chainman  ranks  next  to  the  transitman  in  the 
transit  party,  and  is  directly  in  charge  of  the  rear  chainman, 
stakeman,  and  axmen.  His  duties  are:  to  see  that  the  distances 
are  chained  correctly;  to  see  that  the  stakes  are  driven  on  line, 
that  they  are  driven  straight,  and  that  they  are  marked  cor- 
rectly; to  direct  the  axmen  where  to  cut  in  opening  up  the  line; 
to  set  new  transit  points ;  and  to  direct  the  taking  of  plusses.  The 
head  chainman  has  a  very  important  position,  as  he  regulates 
the  speed  of  the  entire  party.  In  open  country  frequently  the 
locating  engineer  takes  this  position.  The  head  chainman 
carries  the  zero  end  of  the  tape. 

The  Rear  Chainman's  duties  are :  to  hold  his  end  of  the  tape  on 
the  last  stake  driven  while  the  head  chainman  gets  the  distance; 
and  to  take  and  record  all  plusses  which  he  turns  over  to  the 
transitman  at  frequent  intervals. 

The  Rear  Flagman's  duties  are:  to  give  the  transitman  a 
sight  on  the  back  sight  station  whenever  he  signals  for  it;  and  to 
carry  excess  baggage.  The  rear  flagman  should  be  a  wide- 
awake man  with  good  eyesight. 

The  Stakeman's  duties  are:  to  carry  the  stakes;  to  mark  the 
station  numbers  on  the  stakes;  and  to  drive  the  stakes  as  directed 
by  the  head  chainman. 

The  Axmen  do  all  the  necessary  clearing  in  order  that  the 
transit  and  level  parties  may  have  a  clear  path.  They  are 
sometimes  required  to  make  the  stakes. 


RAILROAD    SURVEYS 

26.  THE  LEVEL  PARTY.— The  members  of  the  level  party 
and  their  duties  are  as  follows: 

The  Levelman  is  chief  of  the  level  party  and  ranks  next  to  the 
transitman  in  the  surveying  corps.  His  duties  are:  to  run  profile 
levels  over  the  line  and  to  establish  bench  marks;  and  to  keep 
the  level  notes. 

The  Rodman's  duties  are:  to  hold  the  rod  vertical  upon  the 
ground  at  each  station,  and  at  those  intermediate  points  where 
the  longitudinal  slope  of  the  ground  changes;  and  to  keep  "  peg 
notes  "  as  a  check  on  the  levelman's  computations. 

27.  THE  TOPOGRAPHY  PARTY.— The    members    of    the 
topography  party  and  their  duties  are  as  follows: 

The  Topographer  usually  holds  equal  rank  with  the  levelman 
in  the  surveying  corps.  This  position  is  a  very  important 
one  and  should  be  filled  by  an  experienced  man.  The  duties  of 
the  topographer  are:  to  take  all  data  necessary  for  making  an 
accurate  contour  map  of  a  strip  of  country  sufficiently  wide  to 
enable  the  engineer  to  make  an  intelligent  projected  location; 
and  to  record  these  data  in  such  a  way  that  they  will  be  readily 
understood  by  the  draftsman.  The  topographer  is  assisted 
in  his  work  by  a  rodman  and  a  tapeman. 

28.  THE  LAND-LINE  PARTY.— The    duties   of   this   party 
are :  to  measure  the  angles  which  the  line  makes  with  all  railways, 
highways,  streams,  and  property  lines;  to  tie-in  the  line  to  the 
nearest  government  monuments  so  that  legal  descriptions  may 
be  prepared  of  the  required  right  of  way ;  and  to  secure  the  names 
of  the  property  owners. 

29.  Drafting.     In  addition  to  the  above  field  parties  there  is 
the  field  draftsman,  who  does  his  work  in  camp.     His  duties  are 
to  plat  the  notes  taken  by  all  the  parties  the  previous  day,  which 
necessitates  the  use  of  two  sets  of  field  note-books  or  else  loose-leaf 
note-books.     In  some  cases  the  draftsman,  with  the  help  of  tha 
locating  engineer  and  the  transitman,  plats  the  notes  each  night ; 
and  the  levelman  plats  the  profile  of  the  line  over  which  he  ran 
levels  that  day.     Thus  the  map  is  kept  up  to  date,  and  the 
locating     engineer     can     project     his     location     as     the     line 
advances. 


10  TEXT   BOOK   ON   RAILROAD    SURVEYING 

Methods  of  Making  Preliminary  Surveys 

The  preliminary  survey  is  the  most  expensive  survey,  and  is 
of  primary  importance  since  the  location  depends  directly  upon 
it.  Hence  the  method  of  making  it  should  be  given  considerable 
thought  in  order  that  it  may  be  done  with  accuracy  and  economy. 
There  are  at  present  three  general  methods  used:  (1)  the  transit 
and  tape  method,  (2)  the  transit  and  stadia  method,  and  (3) 
the  plane  table  method. 

30.  The  transit  and  tape  method  is  by  far  the  most  common. 
The  transit  party  runs  the  line  with  transit  and  tape,  the  level 
party  follows  taking  profile  levels,  and  the  topography  party 
follows  the  level  party;  the  land-line  party  may  come  in  any- 
where after  the  transit  party.     This  kind  of  a  survey  will  require 
from  six  to  fifteen  men,  depending  on  whether  the  several  parties 
have  a  separate  personnel  and  on  the  number  of  axmen  required 
to  open  the  line. 

31.  The  transit  and  stadia  method  consists  in  running  the 
line  with  the  transit  and  stadia.     Stakes  are  placed  only  at 
transit  stations,  and  the  elevations  of  these  points  are  determined 
by  transit  and  trigonometric  leveling.     The  location  and  eleva- 
tions of  important  intermediate  points  along  the  line  are  de- 
termined in  a  similar  manner.     While  the  transit  is  at  each 
station,  the  topography  around  that  point  is  taken  with  the  stadia. 
Thus  all  the  needed  data  are  taken  as  the  line  advances.     These 
data  may  be  recorded  in  the  note-book  and  worked  up  later  by 
the  party  draftsman;  but  it  is  much  better  if  the  draftsman  plats 
the  notes  as  the  transitman  takes  them  and  draws  in  the  contours 
while  the  landscape  is  before  him.     The  draftsman  holds  a  very 
important  position  in  this  party  and  should  be  an  expert  in  that 
line  of  work.     A  survey  by  this  method  is  very  accurate  as  re- 
gards the  contours,  which  are  the  most  important  item  on  a 
preliminary  map;  and,  if  good  men  are  employed,  it  is  more 
efficient  than  the  first  method.     This  method  requires  a  transit- 
man, a  draftsman,  two  rodmen,  and  as  many  axmen  as  the 
nature  of  the  country  may  require. 

32.  The  plane  table  method  is  very  similar  to  the  one  just 
described  and  differs  from  it  mainly  in  the  use  of  instruments. 


RAILROAD   SURVEYS  11 

The  plane  table  takes  the  place  of  the  transit,  and  the  plane 
tableman  does  the  drafting.  Owing  to  the  difficulty  in  handling 
and  setting  up  the  plane  table,  it  is  doubtful  whether  this  method 
is  as  efficient  as  the  transit  and  stadia  method;  and  although 
the  services  of  the  draftsman  are  dispensed  with,  yet  the  progress 
is  possibly  not  as  rapid. 

33.  Remarks.     For  long  lines  that  justify  the  employment  of  a 
large  number  of  men  the  transit  and  tape  method  is  probably  the 
most  efficient.     In  other  cases,  however,  the  transit  and  stadia 
method  will  prove  the  more  economical.     The  latter  method 
has  not  been  used  to  any  considerable  extent,  due  to  the  fact  that 
few  engineers  fully   appreciate  the  advantages  of  the  stadia 
method;  but  in  the  few  cases  in  which  it  has  been  tried  it  has 
fully  demonstrated  its  superiority,  particularly  for  open  country. 

34.  Bench  marks.     When  spirit-levels  are  run  over  the  line, 
bench  marks  should  be  established  at  half-mile  intervals,  ap- 
proximately, and  should  be  placed  far  enough  from  the  center 
line  so  that  they  will  not  be  disturbed  during  construction. 
After  construction,  permanent  bench  marks  should  be  established 
on  all  permanent  structures,  such  as  concrete  bridges,  and  at 
every  station  building  along  the  line.     The  common  practice 
of  using  spikes  driven  in  telephone  poles  and  mile  posts  is  a 
very  dangerous  one,  as  these  are  frequently  moved  and  re-set. 
If  the  road  has  few  concrete  or  steel  structures,  then  bench 
marks  may  be  established  on  trestles.     When  these  are  renewed, 
the  elevations  of  the  several  parts  will  rarely  be  changed  more 
than  an  inch;  while  a  bench  mark  on  a  telephone  pole  may  be 
changed  several  feet. 


Note. — For  further  information  on  the  subject  of  Railway  Surveys 
and  the  Economics  of  Railway  Location,  see  A.  M.  Wellington's 
"Economic  Theory  of  the  Location  of  Railways; "  F.  Lavis'  "  Railway 
Location  Surveys  and  Estimates,"  or  W.  L.  Webb's  "Economics  of 
Railroad  Construction." 


CHAPTER  II 

MAPS 

Reconnoissance  Maps 

35.  The  reconnoissance  map  consists  of  a  general  sketch  of 
the  country  which  the  locating  engineer  has  investigated,  and 
shows  the  several  routes  that  are  possible.     Only  controlling 
points,  such  as  towns  to  be  passed  through,  available  stream 
crossings,  saddles  in  the  ridges,  etc.,  are  shown.     If  an  existing 
map  of  the  country  is  available,  it  is  best  to  draw  in  the  routes 
directly  upon  it.     The  notes  taken  on  the  reconnoissance  survey 
regarding  the  geological  formations,  the  cultural  features  of  the 
country,  the  width,  depth,  and  current  of  streams,  etc.,   are 
considered  part  of  the  reconnoissance  map,  and  frequently  have 
great  weight  in  the  choice  of  routes. 

36.  The  reconnoissance  profile  is  made  from  the  elevations  of 
the  controlling  points  and  the  distances  between  them.     Pro- 
files are  made  for  each  of  the  several  routes  and  are  frequently 
placed  on  the  same  sheet,  so  that  a  more  intimate  comparison 
can  be  made.     The  purpose  of  the  reconnoissance  maps  is  to 
eliminate  the  impractical  routes  and  to  determine  which  ones  will 
bear  a  more  detailed  investigation. 

Preliminary  Maps 

37.  The  first  preliminary  maps  show  the  transit  and  stadia  lines 
run,  and  all  railroads,  highways*  and  streams  that  cross  the  lines. 

38.  The  preliminary  profiles  corresponding  to  the  first  pre- 
liminary maps  are  made  from  the  elevations  of  the  traverse 
stations.     From  a  study  of  these  maps  one  route — occasionally 
two — will  appear  superior  to  the  others,  and  the  preliminary 
survey  proper  is  then  made  over  this  route. 

39.  The  preliminary  map  is  generally  made  to  a  scale  of  from 
200  to  400  feet  to  the  inch,  and  is  a  complete  topographic  map  of 

12 


MAPS  13 

a  strip  of  country  from  100  to  1,000  feet  wide.  All  highways, 
railways,  streams,  and  buildings  are  shown.  Frequently  it  is 
desirable  to  show  the  property  lines,  the  names  of  land  owners, 
and  government  monuments.  Also  all  geological  formations 
and  other  features  that  might  affect  the  location  of  the  line  are 
indicated.  Occasionally,  special  features  may  require  an  aux- 
iliary map  to  a  scale  of  50  or  100  feet  to  the  inch  over  short 
stretches  of  the  line. 

40.  The  projected  location  is  then  made  on  the  preliminary  map. 
First  project  that  line  which  will  give  the  best  alinement  and  grades. 
This  will  be  the  best  line  from  the  standpoint  of  operation,  and 
the  one  to  be  used  unless  the  cost  of  construction  is  prohibitive. 
In  this  case  determine  the  line  whose  construction  will  come  with- 
in the  allowable  cost  and  at  the  same  time  keeps  within  the 
allowable  limits  of  curvature  and  grade.     This  can  be  done  only 
by  trial  and  requires  a  large  amount  of  skill  and  judgment. 
In  order  to  determine  the  grades  and  the  amount  of  earthwork 
for  the  several  projected  lines,  it  will  be  necessary  to  construct 
a  profile  for  each  of  them  from  the  contour  map.     In  locating 
grade  lines  on  these  profiles  it  must  be  remembered  (1)  that 
intersecting  railways  must  be  crossed  either  at  grade  or  at  a 
clearance  distance  above  or  below  grade,    (2)   that  highways 
can  be  raised  or  lowered  within  certain  limits,  and   (3)  that 
streams  must  be  crossed  a  safe  distance  above  high-water  mark. 

After  satisfactory  tangents  have  been  projected,  they  are 
connected  by  curves  which  most  nearly  conform  to  the  contour 
of  the  ground,  and  at  the  same  time  keep  within  the  maximum 
curvature.  If  the  curves  are  to  be  spiraled,  allowance  must  be 
made  at  this  time. 

41.  Location  notes  are  then  made  from  the  accepted  projected 
location.     The   bearings   of   tangents,   the   plusses   of   the   be- 
ginnings and  ends  of  all  curves,  the  central  angles  of  all  curves, 
and  the  degrees  of  curve  are  scaled  off  the  map  and  recorded. 
These  condensed  notes  are  used  in  making  the  final  location. 

Location  Map 

42.  The  location  map  is  usually  made  to  the  same  scale  as  the 
preliminary  map,  and  shows  all  railways,  highways,  streams, 
and  property  lines  that  cross  the  line,  together  with  their  plusses; 


14  TEXT   BOOK   ON   RAILROAD   SURVEYING 

the  names  of  the  property  owners  and  the  amount  of  right  of  way 
required  from  each;  the  government  subdivision  lines  and  the 
numbers  of  sections,  etc.;  the  distances  from  the  line  to  govern- 
ment monuments;  the  boundaries  of  each  field  through  which  the 
line  passes;  the  location  and  size  of  all  openings;  and  any  other 
data  that  will  be  of  use  to  the  construction  engineer  or  to  the 
right  of  way  agent. 

The  location  map  can  often  be  made  by  adding  the  necessary 
data  to  the  preliminary  map,  and  then  making  a  new  tracing, 
omitting  the  contours  and  such  other  features  that  are  not  de- 
sired on  the  final  map. 

43.  The  location  profile  is  made  from  the  profile  levels  which 
were  run  over  the  located  line,  and  shows:  the  ground  line;  the 
grade  line,  including  the  per  cent,  of  grade  of  the  different  por- 
tions and  the  elevations  of  all  points  where  the  grade  changes 
and  all  points  on  vertical  curves;  the  location  and  dimensions  of 
all  openings  in  the  embankment ;  the  plusses  of  railways,  highways, 
streams,  and  openings  in  the  embankment ;  the  elevations  of  the 
top  of  rail  of  all  intersecting  railroads,  and  the  high-water  mark 
of  streams;  the  descriptions  and  elevations  of  all  bench  marks;  a 
rectified  alinement  map  at  the  bottom  of  the  profile  sheet,  con- 
taining practically  all  the  data  that  are  shown  on  the  location 
map;  the  distribution  diagram,  which  is  drawn  between  the 
profile  proper  and  the  alinement;  on  the  profile  proper  the  econom- 
ical movement  of  the  earth  from  the  cuts  into  the  fills;  and  the 
amount  of  excavation,  overhaul,  and  borrow  at  all  points  along 
the  line.     In  fact,  the  location  profile  contains  practically  all  the 
information  needed  by  the  resident  and  construction  engineers. 
It  is  used  by  them  in  staking  out  all  the  construction  work,  with 
the  exception  of  large  bridges  and  trestles,  for  which  separate 
plans  are  made. 

44.  Additional  profiles  are  made  for  any  change  in  the  eleva- 
tions of  highways  and  railways.     These  profiles  are  short  and 
are  sometimes  (and  preferably)  placed  on  the  location  profile 
opposite  the  points  where  the  changes  are  made. 

Right  of  Way  Maps 

Right  of  way  maps  are  of  two  kinds:  (1)  legal  maps  and  (2) 
maintenance  of  way  maps. 


MAPS  15 

45.  Legal  right  of  way  maps  are  made  on  sheets  which  are  the 
same  size  as  the  sheets  on  which  the  conveyance  deeds  are  drawn 
up  (usually  8J  x  13).     A  separate  map  is  made  for  each  de- 
scription, and  a  blue-print  is  attached  to  the  deed  and  becomes  a 
part  of  it  and  is  recorded  along  with  the  deed.     The  railroad 
company  preserves  its  copies  of  the  deeds  along  with  the  right 
of  way  maps  in  the  form  of  a  book,  which  is  known  as  the  right 
of  way  book. 

46.  Maintenance  of  way  right  of  way  maps  are  made  from  the 
location  map  and  a  separate 'sheet  (about  8J  x  10)  is  used  for 
each  section  (i.e.,  government  section)  or  for  each  mile  of  track. 
These  sheets  are  bound  into  a  book  and  are  used  by  the  main- 
tenance of  way  department. 

Construction  Maps 

47.  Station  maps  are  usually  made  to  a  scale  of  50  or  100  feet 
to  the  inch,  and  show  the  proposed  buildings  and  tracks  at  each 
station.     Any  changes  in  the  plans  which  are  made  during  con- 
struction are  recorded  on  the  map.     Some  railroads  have  stand- 
ard plans  for  station  layouts,  and  these  are  followed  as  closely 
as  conditions  will  permit. 

48.  Progress  profiles.     Each  month   after  the  monthly  es- 
timate surveys  have  been  made,  the  amount  of  grading  done 
during  the  month  is  shown  graphically  on  the  location  profile  in 
colored  pencil.     Each  month  a  different  color  is  used,  so  that 
the  chief  engineer  and  other  railroad  officials  can  see  at  a  glance 
the  progress  of  the  construction  work  and  can  compare  the 
amount  done  each  month  with  that  of  former  months. 

49.  Progress  photographs.     For  the  same  reason  photographs 
should  be  taken  each  month  or  oftener  of  the  various  structures 
under  construction,  such  as  concrete  arches,  bridges,  trestles, 
station   buildings,   etc.     Photographs   are   excellent   auxiliaries 
to  a  written  report. 


CHAPTER  III 

DISTANCE,  CURVATURE,  AND  GRADES 

IN  the  location  of  a  railroad  there  are  three  factors  that  make 
for  success  or  failure  from  a  financial  standpoint,  namely,  dis- 
tance, curvature,  and  grades.  The  best  location  is  that  in  which 
each  of  these  is  a  minimum.  This  would  be  when  the  line  be- 
tween terminals  was  perfectly  straight,  and  on  a  uniform  grade. 
This  ideal  line  involves  a  maximum  expenditure  for  construction 
and  a  minimum  outlay  for  operation.  Such  a  line,  of  course, 
is  a  financial  if  not  a  physical  impossibility.  It  must  be  re- 
membered that  the  first  cost  is  a  fixed  amount,  while  the  cost  of 
operation  continues  as  long  as  the  road  exists;  and  in  time  the 
additional  cost  of  operating  a  cheaply  located  road  will  more 
than  offset  the  amount  saved  in  the  construction.  This  state- 
ment is  borne  out  by  the  fact  that  all  the  large  railroads  are 
spending  huge  amounts  yearly  for  the  reduction  of  curvature 
and  grades,  in  order  that  operating  expenses  may  be  reduced. 
The  locating  engineer  must  take  all  of  these  things  into  con- 
sideration in  making  his  location.  Of  the  three  named  items, 
grades  is  by  far  the  most  important,  while  distance  is  the  least 
important. 

Distance 

50.  Additional  distance  in  the  length  of  a  line  affects  the 
initial  cost  of  construction,  the  cost  of  maintenance,  and  the  cost 
of  operation.  The  cost  of  construction  and  maintenance  is 
proportional  to  the  distance  and  may  be  readily  determined. 
The  cost  of  operation  does  not  vary  directly  as  the  distance. 
The  cost  of  operating  a  small  additional  distance  will  be  only 
about  one-third  as  much  per  train-mile  as  the  average  cost  over 
tne  whole  line.  When  the  additional  distance  becomes  several 
miles,  the  cost  is  increased  to  about  one-half  of  the  average  cost 
per  train-mile. 

16 


DISTANCE,    CURVATURE,    AND    GRADES  17 

Curvature 

51.  Curvature  affects  the  cost  of  construction,  the  cost  of 
maintenance,  and  the  cost  of  operation. 

The  cost  of  construction  is  increased  since  curvature  means 
an  increase  in  the  length  of  the  line;  and  also  if  structures  such  as 
trestles  and  bridges  come  on  curves  their  cost  is  largely  in- 
creased thereby. 

The  cost  of  maintenance  is  increased  since  curved  track 
requires  more  care  in  alinement  and  surface  than  straight  track, 
and  also  there  is  greater  wear  on  both  the  track  and  the  rolling 
stock  due  to  the  curvature. 

The  cost  of  operation  is  increased  because  more  power  is  re- 
quired on  curves  than  on  straight  track,  and  frequently  a  sharp 
curve  will  limit  the  weight  of  train  that  can  be  operated  over  the 
line.  Also,  sharp  curves  limit  the  speed  of  passenger  trains, 
which  may  be  an  important  item  where  there  is  a  competing 
line.  It  has  been  determined  by  experiment  that  the  additional 
tractive  force  required  to  draw  a  train  around  a  one-degree  curve 
is  the  same  as  that  required  by  a  grade  of  from  0.03  to  0.05  per 
cent.  This  resistance  does  not  vary  directly  as  the  degree  of 
curve,  but  decreases  as  the  degree  of  curve  increases.  For  curves 
up  to  about  8  degrees,  however,  the  curvature  resistance  is 
usually  taken  as  0.04  per  cent,  per  degree  of  curve;  for  curves 
from  8  to  15  degrees,  0.03  per  cent,  per  degree;  and  for  curves 
above  15  degrees,  0.02  per  cent,  per  degree.  The  resistance  to  a 
train  in  starting  is  about  double  that  given  above,  and  must  be 
considered  if  a  stopping  place  comes  on  a  curve.  In  general, 
curve  resistance  is  not  serious  until  it  becomes  so  great  that 
when  added  to  grade  resistance  the  total  resistance  limits  the 
weight  of  train  that  can  be  hauled  over  the  line.  Therefore  all 
the  steeper  grades  should  be  compensated  for  curvature. 

52.  Choice  of  curvature.     The  total  resistance  offered  by  a 
curve  depends  directly  upon  the  central  angle,  and  is  practically 
independent  of  the  radius.     Hence  the  important  point  is  to 
keep  the  amount  of  curvature  a  minimum,  and  after  the  tangents 
have  been  decided  upon  the  problem  is  to  make  the  rate  of 
curvature  as  uniform  as  the  topography  will  permit.     Little  is 
gained  by  having  a  few  flat  curves  and  a  number  of  sharp  ones. 
The  proof  of  this  statement  is  in  the  fact  that  in  the  vast  amount 


18  TEXT   BOOK    ON    RAILROAD    SURVEYING 

of  re-alinement  work  that  the  railroads  have  been  doing  in  the 
past  few  years  there  are  very  few,  if  any,  cases  where  an  increase 
in  radius  is  the  only  benefit  derived  from  the  improvement. 
The  object  of  all  this  work  has  been  to  reduce  the  total  amount 
of  central  angles  in  curves  and  at  the  same  time  to  reduce  the 
maximum  rate  of  curvature. 

Grades 

Grades  may  be  classified  as  (1)  minor  grades,  (2)  ruling  grades, 
and  (3)  pusher  grades. 

53.  Minor  grades  are  those  which  do  not  limit  the  weight  of 
train  which  can  be  hauled  over  the  line  by  one  engine.     Such 
grades  are  of  three  kinds  according  to  their  effect  on  the  per- 
formance of  the  engine. 

First,  there  are  those  grades  whose  drop  (vertical  height)  is 
sufficiently  small  that  a  locomotive  can  operate  without  shutting 
off  steam,  or,  in  other  words,  will  not  reach  a  dangerous  speed  due 
to  the  down-grade  while  exerting  a  constant  pull  on  the  train. 
The  effect  of  these  grades  on  train  operation  is  negligible,  both  as 
regards  the  effort  required  of  the  engine  and  the  time  required 
for  the  train  to  traverse  a  given  distance,  since  the  kinetic  energy 
acquired  in  descending  one  grade  is  utilized  in  ascending  the 
opposite  grade. 

The  second  kind  of  minor  grades  consists  of  those  whose  drop  is 
so  great  that  steam  must  be  shut  off  to  prevent  the  train  from 
acquiring  a  dangerous  velocity,  and  hence  the  effort  of  the 
engine  must  be  increased  on  the  opposite  up-grade.  Therefore 
there  is  a  loss  of  power,  since  the  engine  is  working  intermit- 
tently and  not  at  its  greatest  efficiency. 

The  third  kind  of  minor  grades  consists  of  those  which  require 
the  use  of  brakes  in  descending  them.  This  causes  an  enormous 
loss  of  energy  because  part  of  the  kinetic  energy  gained  by  the 
descent  is  absorbed  by  the  brakes,  and  hence  is  not  available  on 
the  next  up-grade.  Further,  the  engine  is  working  at  a  dis- 
advantage, and  there  is  considerable  wear  on  equipment  due  to 
the  action  of  the  brakes. 

54.  Ruling  grades  are  those  which  limit  the  weight  of  the 
train  which  can  be  hauled  by  one  engine.     Cars  can  not  be 
picked  up  or  dropped  off  along  the  line  to  make  up  a  train  in 


DISTANCE,    CURVATURE,    AND    GRADES  19 

accordance  with  the  grades  met  with,  and  hence  a  train  must 
run  through  from  one  terminal  of  a  division  to  the  other.  There- 
fore, excluding  pusher  grades,  and  such  short  steep  grades  which 
can  always  be  operated  by  momentum,  the  maximum  grade  is 
the  ruling  grade  on  that  division. 

55.  A  pusher  grade  is  one  which  is  so  steep  that  one  or  more 
extra  engines  are  required  to  haul  the  train  which  one  engine 
can  handle  on  the  remainder  of  the  division.     It  will  nearly 
always  happen  that  some  grades  on  each  division  will  be  con- 
siderably greater  than  the  majority  of  the  grades,  and  unless 
these  can  be  reduced  to  the  general  average  at  a  reasonable 
cost  or  a  pusher  engine  is  used,  they  limit  the  weight  of  train 
over  the  division  and  become  the  ruling  grades. 

A  pusher  grade  adds  enormously  to  the  cost  of  operation  since 
the  auxiliary  engines  must  be  maintained;  and,  further,  they  pass 
twice  over  the  line  for  each  train  handled,  and  hence  do  not 
operate  at  maximum  efficiency.  Obviously,  therefore,  a  pusher 
grade  must  be  considerably  in  excess  of  the  ruling  grade  on  the 
remainder  of  the  division  before  its  extra  cost  of  operation  would 
be  justifiable. 

56.  Choice  of  grades.     The  total  energy  available  to  carry 
a  moving  train  up  a  grade  is  the  sum  of  the  kinetic  energy  of  the 
train  due  to  its  velocity  (momentum)  and  that  developed  by  the 
engine.     The  maximum  amount  of  energy  which  the  engine  can 
develop  is  a  fixed  quantity,  but  the  amount  of  kinetic  energy 
which  can  be  utilized  depends  on  the  initial  speed  of  the  train. 

A  given  engine  can  just  pull  a  train  of  given  weight  up  such 
a  grade  that  the  sum  of  the  frictional  resistances  and  the  grade 
resistances  (including  curvature  resistances)  equals  the  total 
force  exerted  by  the  engine.  Or,  conversely,  the  weight  of  train 
which  a  given  engine  can  pull  up  a  given  grade  is  such  that  the 
sum  of  frictional  and  grade  resistances  again  equals  the  effort 
of  the  engine.  The  frictional  resistances  are  practically  constant 
at  about  8  pounds  per  ton  for  freight  trains,  while  the  grade 
resistance  is  20  pounds  per  ton  for  each  per  cent,  of  grade.  The 
length  of  such  a  grade  is  limited  only  by  the  endurance  of  the 
engine. 

If  the  train  approaches  this  grade  at  considerable  speed,  its 
kinetic  energy  may  be  utilized  to  overcome  additional  grade 


20  TEXT   BOOK   ON   RAILROAD    SURVEYING 

resistance,  hence  the  engine  can  pull  the  train  up  a  much  steeper 
grade  to  the  point  where  the  additional  vertical  rise  has  absorbed 
all  the  kinetic  energy.  If  the  end  of  the  grade  is  reached  first, 
the  train  will  pass  over  easily,  suffering  only  a  reduction  in  speed; 
if  the  grade  extends  further  than  this  point,  the  train  will  be 
stalled. 

Thus  it  is  seen  that  for  minor  grades  it  is  more  the  vertical 
height  than  the  rate  of  slope  which  affects  the  operation.  And 
also  the  ruling  grade — the  one  which  limits  the  train  weight — 
may  not  be  the  one  of  steepest  elope,  provided  that  the  heaviest 
trains  can  always  approach  it  at  sufficient  speed  to  reach  its  top 
with  the  aid  of  momentum. 

Therefore  if  a  grade  line  can  be  established  with  alternate  de- 
scents and  ascents  of  such  small  amount  that  the  train  can 
always  make  use  of  its  momentum,  the  rate  of  slope  is  of  small 
importance.  But,  if  the  slopes  are  so  long  that  momentum 
can  not  be  utilized  all  the  way,  or  the  grade  must  be  entered  at 
very  low  speed,  the  rate  of  slope  limits  the  train  weight. 

The  first  step  in  projecting  a  grade  line  on  a  profile  is  to 
determine  approximately  the  ruling  grade.  The  lowest  value  of 
the  ruling  grade  is  that  of  a  uniform  slope  between  terminals. 
The  maximum  value  depends  on  the  type  and  weight  of  locomo- 
tives, the  weight  of  trains,  etc.,  and  can  never  be  exactly  de- 
termined, and  therefore  it  is  more  usual  to  simply  choose  a 
maximum  or  ruling  grade  which  will  probably  fit  the  territory 
through  which  the  line  runs.  In  assuming  this  maximum  value, 
due  consideration  should  be  given  to  the  direction  of  heaviest 
traffic,  and  easier  grades  secured  if  possible  for  trains  in  this 
direction. 

A  grade  line  is  then  laid  out  on  the  profile,  keeping  below  this 
maximum  if  possible  in  such  a  way  that  the  fills  will  balance  the 
cuts  and  the  total  amount  of  earthwork  will  be  kept  as  low  as 
possible  in  order  to  reduce  the  cost  of  construction.  The  laying 
out  of  this  line  will  require  several  trials,  and  each  trial  should  be 
carefully  studied  as  to  its  effect  on  operation  as  well  as  on  first 
cost.  After  such  a  line  has  been  laid  out  it  may  be  found  that 
the  assumed  maximum  grade  has  been  exceeded  at  some  point, 
and  study  must  be  given  as  to  whether  it  is  possible  to  reduce 
this  grade  or  whether  it  is  justifiable  to  perhaps  increase  it  suf- 
ficiently to  be  operated  as  a  pusher  grade.  Or  it  may  be  found 
that  only  one  grade  approaches  the  chosen  maximum,  and  again 


DISTANCE,    CURVATURE,    AND    GRADES  21 

the  line  must  be  studied  to  determine  if  it  is  possible  to  reduce 
this  grade  to  the  general  average.  For  example,  the  maximum 
grade  on  a  division  is  found  to  be  1  per  cent.,  while  there  are  no 
others  over  0.8  per  cent.,  and  possibly  only  two  of  these,  the  next 
lower  being  0.5  per  cent.  The  problem  then  is  to  decide  the 
advisability  of  reducing  the  1  per  cent,  grade  to  0.8  per  cent., 
and  then  possibly  the  three  0.8  per  cent,  grades  to  the  average 
of  0.5  per  cent.  And  this  process  of  reducing  the  ruling  grade 
could  be  continued  until  the  additional  cost  of  construction 
equals  the  sum  which  can  profitably  be  spent  for  the  purpose  of 
reducing  the  ruling  grade. 

57.  VERTICAL  CURVES. — The  intersections  of  the  several 
portions  of  the  grade  line  must  be  rounded  off  to  avoid  undue 
stress  in  the  drawbars.  The  curve  used  for  this  purpose  is  a  parab- 
ola because  of  its  convenient  characteristics.  Such  curves  are 
termed  vertical  curves  to  distinguish  them  from  the  curves  in  the 
alinement. 

The  length  of  vertical  curves  depends  on  the  total  change  in 
the  grade  which  is  the  algebraic  difference  of  the  intersecting 
grades,  Gi  and  G?.,  and  on  the  rate  of  change  of  the  vertical 
curve.  Ascending  grades  are  plus,  and  descending  grades  are 
minus. 

The  rate  of  change  per  station,  r,  of  the  vertical  curve  as  rec- 
ommended by  the  American  Railway  Engineering  Association 
is  0.1  foot  for  summits  and  0.05  foot  for  sags  for  first-class  rail- 
ways; and  double  these  amounts  for  second-class  and  electric 
railways.  It  is  to  be  noted,  however,  that  a  large  number  of 
first-class  roads  use  the  second-class  rating.  Since  the  change 
at  the  ends  of  the  curve  is  from  tangent  to  chord  instead  of  from 
chord  to  chord,  the  rate  of  change  of  the  first  and  last  stations 
on  the  curve  is  just  one-half  the  rate  used  on  the  remainder  of 
the  curve. 

In  solving  a  vertical  curve  problem,  the  first  step  is  to  find 
the  length  of  the  curve.  This  is  done  by  dividing  the  algebraic 
difference  of  the  grades  by  the  assumed  rate  of  change  per  station. 
This  will  usually  give  an  odd  length  which  is  inconvenient  to 
handle,  and  therefore  the  next  higher  even  number  of  stations  is 
chosen  as  the  length  of  the  curve. 

The  second  step  is  to  divide  the  algebraic  difference  of  the 
grades  by  the  length  of  curve  as  found  in  step  one.  The  quotient 


22  TEXT   BOOK   ON   RAILROAD    SURVEYING 

will  be  the  rate  of  change  per  station  which  is  to  be  actually 
used. 

The  third  step  is  to  find  the  station  numbers  and  elevations 
of  the  T.C.  and  C.T.  (beginning  and  end  of  curve). 

The  fourth  step  is  to  find  the  elevations  of  the  intermediate 
stations  on  the  curve.  This  is  done  by  starting  at  the  T.C. 
and  adding  (or  subtracting  as  the  case  may  be)  the  changing 
grades  successively  until  the  C.T.  is  reached.  The  grade  from 
the  T.C.  to  the  first  station  will  be  GI  —  \  r,  from  the  second  to  the 
third  station  it  will  be  GI — -f  r,  from  the  third  to  the  fourth  it 
will  be  GI — •§•  r,  and  so  on  till  the  C.  T.  is  reached.  The  elevation 
thus  found  for  the  C.  T.  should  check  with  that  determined  from 
step  three. 

Example.  A  +0.8  per  cent,  grade  meets  a  —0.6  per  cent, 
grade  at  station  30  +  00,  whose  elevation  is  750.50.  It  is  de- 
sired to  connect  these  grades  by  a  vertical  curve  whose  rate  of 
change  per  station  is  approximately  0.2  foot. 


GI  —  G2  (algebraically) 
Length  of  curve  =  — 2  v    *  '    .     .     . 

f 

L,W.s4,-,7 


r  -     -    -    CD 


Making  L  =  8, 

r  =  -^  =  0.175 

o 

Taking  everything  into  consideration  it  is  better  to  make 
L  =  800  feet  (400  feet  each  side  of  apex),  and  to  use  a  value  of 
r  =  0.175  than  it  is  to  make  L  =  700  ft.  and  use  a  value  of 
r  =  0.2. 

Sta.  of  T.C.  =  30  -  4  =  26  +  00. 

Sta.  of  C.T.  =  30  +  4  =  34  +  00. 

Elev.  of  T.C.  =  Elev.  P.I.  -  \LGi  =  750.50  -  3.20  =  747.30 

Elev.  of  C.T.  =  Elev.  P.I.  -  \LGi  =  750.50  -  2.40  =  748.10 

Elevation  Station  26  =  747.30  =  747.30 

"       27  =  747.30  +  (0.8  -  0.0875) 

=  747.30  +  0.7125  =  748.01 


DISTANCE,    CURVATURE,   AND   GRADES  23 

Elevation  Station  28  =  748.01  +  (0.7125  -  0.175)  = 

748.01  +  0.5375  =  748.55 
"  "      29  =  748.55  +  (0.5375  -  0.175)  = 

748.55  -f  0.3625  =  748.91 
"      30  =  748.91  -f  (0.3625  -  0.175)  = 

748.91  +0.1875  =  749.10 
"       31  =  749.10  +  (0.1875  -  0.175)  = 

749.10  -h  0.0125  =  749.11 
"              "       32  =  749.11  +  (0.0125  -  0.175)  = 

749.11  -  0.1625  =  748.95 
"       33  =  748.95  +  (-0.1625  -  0.175)  = 

748.95  -  0.3375  =  748.61 
"  "       34  =  748.61  +  (-0.3375  -  0.175)  = 

748.61  -  0.5125  =  748.10 

It  should  be  noted  that  the  highest  point  on  the  vertical  curve 
comes  at  Sta.  31  instead  of  Sta.  30. 

The  elevation  of  each  station  on  the  vertical  curve  is  shown 
on  the  location  profile,  and  is  used  in  setting  slope  stakes.  If 
the  distance  from  the  sub-grade  to  the  top  of  rail  (from  1.5  to 
2.0ft.)  is  added  to  the  above  elevations  the  results  are  the  eleva- 
tions of  the  final  top  of  rail.  These  elevations  are  used  after  the 
grading  has  been  completed  for  setting  grade  stakes  for  sur- 
facing the  track. 


Note. — For  a  further  discussion  on  the  Economics  of  Railway  Loca- 
tion, the  student  is  referred  to  Wellington's  "Economic  Theory  of  the 
Location  of  Railways,"  or  Webb's  ••  Economics  of  Railroad  Construc- 
tion." 


CHAPTER  IV 
CURVES 

Part  I.    Circular  Curves 

RAILWAY  alinement  consists  of 

1.  Tangent,  or  straight  track. 

2.  Curves,  which  unite  the  tangents. 

Curves  are  of  two  kinds;  (1)  arcs  of  circles,  and  (2)  arcs  of 
spirals.  The  circle  is  employed  for  the  body  of  all  curves  since 
it  is  more  easily  located  with  the  transit  and  tape  than  any  other 
form  of  curve.  Spirals  are  used  for  easements  at  the  ends  of  the 
circular  arcs. 

Circular  curves  are  classified  as  Simple,  Compound,  and 
Reversed. 

Simple  Curves 

58.  FUNCTIONS. 

A  simple  curve  is  an  arc  of  a  circle  which  unites  two  tangents 


differing  in  direction.     The   functions  of  a  simple   curve  are 
shown  in  Fig.  2. 

24 


CURVES  25 

Point   of  Intersection  —  P.I.  —  is   the   point   where   the   two 
tangents  intersect. 

Tangent  to  Curve  —  T.C.  —  is  the  end  of  the  tangent  and  the 
beginning  of  the  curve. 

Curve  to  Tangent  —  C.T.  —  is  the  end  of  the  curve  and  the 
beginning  of  the  tangent. 

Intersection  Angle  —  7  —  is  the  deflection  angle  between  the  two 
tangents,  and  is  equal  to  the  angle  at  the  center. 

The  Radius  of  the  curve  is  denoted  by  R. 

Tangent  Distance  —  T  —  is  the  distance  from  the  T.C.  or  the 
C.T.  to  the  P.I. 

T  =  R  tan  \I.        .•  .....     (2) 

Long  Chord—  L.C.—  is  the  chord  from  the  T.C.  to  the  C.T. 

L.C.  =  2fl  sin  i/  .......     (3) 

Middle  Ordinate  —  M  —  is  the  ordinate  to  the  curve  from  the 
middle  of  the  Long  Chord. 

M  =  R  -  R  cos  i/  =  R  vers  JJ        ...     (4) 

External  Distance  —  E  —  is  the  distance  from  the  middle  of  the 
curve  to  the  P.I. 

E  =  ~jR 


Degree  of  Curve  —  D  —  is  the  angle  at  the  center  subtended  by 
one  100-ft.  chord,  two  50-ft.  chords,  or  four  25-ft.  chords,  de- 
pending upon  the  length  of  the  radius.  The  intention  is  to  make 
the  difference  between  the  length  of  the  chord  and  the  arc  so 
small  that  it  may  be  neglected. 

Length  of  Curve  —  L  —  is  the  distance  along  the  curve  between 
the  T.C.  and  the  C.T. 

L  (in  stations)    =  -yr     .....     (6) 

Deflection  Angle  —  \D  —  From  geometry  the  angle  between  two 
chords,  or  a  chord  and  a  tangent,  intersecting  on  the  circum- 


26  TEXT   BOOK   ON   HAILROAD    SURVEYING 

ference  of  a  circle  is  measured  by  one-half  the  intercepted  arc. 
For  a  chord  100  feet  in  length  this  angle  is  called  the  deflection 
angle,  and  therefore  is  JD. 

Subdeflection  Angle — d — is  a  similar  angle  for  a  chord  of  less 
than  100  feet. 

If  C  is  the  length  of  the  chord  in  feet, 

Q 
d  (in  degrees)    =^3^5      •     •     •     •     •     (7) 

d  (in  minutes)  =  0.3CD       (8) 

Total  Deflection  Angle  is  the  angle  at  the  T.C.  between  the 
tangent  and  a  chord  to  any  point  on  the  curve. 

Total  deflection  angles  are  taken  to  the  nearest  0.5  minute. 

59.  The  relation  between  R  and  D.  If  the  degree  of  curve  be 
defined  as  the  angle  at  the  center  subtended  by  a  100-ft.  arc, 
an  exact  relation  between  R  and  D  may  be  found,  as  follows. 
The  circumference  of  a  complete  circle  in  terms  of  the  radius  is 

O(\(\ 

2  nR'}  and  in  terms  of  the  degree  of  curve  is  100  -~- ,  hence 

100  X  360      5729.58 
R=      2*D       =-D~ ^ 

From  (9)  it  is  evident  that  R  varies  inversely  as  D.  Therefore 
knowing  the  radius  of  a  1  degree  curve,  the  radius  of  any  other 
degree  of  curve  can  be  found  by  simple  proportion. 

Since  the  chord  is  used  instead  of  the  arc,  the  value  of  R  from 


FIG.  3. 


(9)  can  not  be  used,  and  a  new  value  of  R  must  be  found  which 
will  agree  with  the  chord-definition  of  the  degree  of  curve. 


CURVES  27 

In  Fig.  3,  R  sin  JD  =  |C 

Whence  R  =  -^75  =  iC  cosec  £D      ....     (10) 

Dill    2  •*-' 

When  C  is  100  ft.,        #  =  50  cosec  il> (11) 

The  difference  in  the  values  of  R  as  computed  from  (9)  and 
(11)  becomes  about  0.5  ft.  when  D  is  7  degrees.  Since  a  difference 
greater  than  this  is  objectionable,  curves  above  7  degrees  are 
staked  out  with  50-ft.  chords.  When  C  is  50  ft., 

R  =  25  cosec  JD (12) 

When  D  becomes  14  degrees,  the  difference  in  R  as  computed 
from  (9)  and  (12)  again  become  objectionable,  and  25-ft.  chords 
are  used.  When  C  is  25  ft., 

R  =  12.5  cosec  I D (13) 

When  D  becomes  28  degrees,  10-ft.  chords  should  be  used. 
Table  5  of  radii  was  computed  from  (11),  (12*),  and  (13). 
It  is  customary  to  consider  the  radius  of  a  1  degree  curve  as 
5730  ft.,  and  to  assume  that  R  varies  inversely  as  D.    Hence 

R  =  —p-  (approx.) (14) 

The  difference  between  R  as  given  in  Table  5,  and  as  computed 
from  (14)  is  so  small  that  it  may  be  neglected. 

By  substituting  R  in  terms  of  D  in  (2)  to  (6)  it  is  seen  that 
the  various  functions  of  a  curve  for  any  given  value  of  7  are 
approximately  inversely  proportional  to  D.  Therefore,  if  these 
functions  are  known  for  any  degree  of  curve,  they  can  be  found 
for  any  other  degree  of  curve  by  simple  proportion.  Table 
6  gives  the  functions  of  a  1  degree  curve  for  intersection  angles 
from  0  degree  to  100  degrees. 

60.  FIELD  WORK. 

In  laying  out  a  curve  successive  points  are  located  by  a  meas- 
urement from  the  preceding  point  and  by  line.  The  line  is 


28  TEXT   BOOK   ON   RAILROAD   SURVEYING 

not  the  same  for  all  points  and  is  determined  by  the  total  de- 
flection angle  calculated  for  each  point.  Thus  in  Fig.  4,  point 
1  is  located  by  the  chord  T.C. — 1  and  the  total  deflection  angle 
A;  point  2  by  the  chord  1-2  and  the  total  deflection  angle  B] 
and  point  3  by  the  chord  2-3  and  the  angle  C. 


FIG.  4. 


Stakes  are  usually  placed  100  feet  apart  except  on  sharp 
curves  where  the  interval  is  reduced  to  50  or  25  feet.  However, 
the  stationing  is  continued  unbroken  around  the  curve.  The 
T.C.  and  the  C.T.  will  therefore  usually  fall  at  a  plus  station, 
hence  there  will  usually  be  a  subdeflection  angle  at  each  end  of 
the  curVe. 

The  first  step  in  the  field  work  is  to  determine  the  station 
numbers  of  the  T.C.  and  the  C.  T.  and  to  compute  the  total  de- 
flection angles  for  the  entire  curve.  In  Fig.  4,  T.C. — 1  and  4- 
C.T.  are  odd  distances,  and  1-2,  2-3,  and  3-4  are  full  stations. 
Then  A  is  the  subdeflection  angle  (di)  for  the  chord  T.C.-l, 
and  is  computed  by  Eq.  (8). 

A  =  di 

Then     B  =  A  +  $D  =  di  +  \D 

And       C  =  B  +  $D  =  d,  +  $D  +  |D 

E  =  C  +  *£>  =  rfi  +  $D  +  $D  +  iD 

F  =  E  +  d2    =  di  +  \D  -h  JD  +  iD  +  &  =  \I 

The  total  deflection  angles  are  thus  computed  by  successive 
additions,  and  the  entire  series  of  computations  is  checked  if 
the  last  value  is  \I.  If  stakes  are  placed  50  feet  apart  the  in- 
crements are  \D,  and  if  25  feet  apart  they  are  \ D,  instead  of 
JD  as  above.  The  form  of  notes  is  shown  in  Fig.  5. 


CURVES  29 

On  the  ground  the  T.C.  and  the  C.T.  are  located  by  measuring 
the  tangent  distance  T  from  the  P. 7.  The  curve  can  then  be  run 
in  by  means  of  angles  and  distances  as  explained  above. 

Since  in  turning  off  an  angle  with  a  transit,  an  error  as  large 
as  0.5  minute  may  easily  be  made— which  amounts  to  0.15  of 
a  foot  at  1000  feet  from  the  transit — the  length  of  sight  on  a 
curve  should  never  exceed  1000  feet. 

It  has  been  found  by  experience  that  if  the  angle  between  the 
tape  and  the  line  of  sight  is  more  than  about  30  degrees,  the 
location  of  the  point  is  inaccurate.  Therefore  the  angle  between 
the  line  of  sight  and  the  tangent  at  the  transit  station  should 
never  be  more  than  30  degrees,  i.e.,  the  product  of  the  length 
of  sight  in  stations  and  |D  should  be  less  than  30  degrees. 

It  is  good  practice  in  any  case  to  run  a  portion  of  the  curve 
from  the  C.T.,  since  the  errors  of  surveying  can  be  adjusted  more 
satisfactorily  on  the  curve  than  at  the  C.  T.  There  should  be  no 
more  error,  however,  in  either  line  or  distance  than  is  permissible 
in  good  chaining  under  the  particular  conditions. 

If  the  entire  curve  can  not  be  run  in  from  the  T.C.  and  the 
C.T.,  one  or  more  intermediate  set-ups  on  the  curve  will  be 
necessary.  There  are,  therefore,  three  possible  positions  of  the 
transit  in  running  in  curves,  viz.j  the  T.C.,  any  intermediate 
point,  and  the  C.T.  The  curve  notes  as  computed  above  are 
used  in  all  three  cases,  as  follows: 

At  the  T.C.  Orient  the  transit  by  a  sight  along  the  tangent 
with  the  plates  set  at  zero.  Turn  off  the  total  deflection  angle 
of  each  station,  successively,  and  chain  the  corresponding  dis- 
tances between  them. 

At  any  intermediate  point.  Orient  the  transit  by  a  back- 
sight on  the  last  transit  station  with  the  telescope  inverted  and  the 
plates  set  at  the  total  deflection  angle  of  that  station  as  recorded 
in  the  notes.  To  continue  the  curve,  plunge  the  telescope  * 
and  set  the  plates  at  the  total  deflection  angles  of  the  succeeding 
stations  and  measure  the  corresponding  distances  between 
them. 

At  the  C.T.  Orient  the  transit  by  a  sight  along  the  tangent 
with  the  telescope  normal  and  the  plates  set  at  the  total  de- 
flection angle  of  the  C.T.,  i.e.,  J7.  To  run  in  the  curve,  set  the 

*  The  line  of  collimation  must  be  in  adjustment. 


TRANSIT  NOTES  FOR  LINE  L 

STATION 

.    ALINE- 
MENT 

TOTAL 
DEFL. 
ANGLE 

CALC. 
BEAR. 

MAG. 
BEAR. 

REMARKS 

-{-  55.8 

O  C.T. 

35°-4l' 

-f  — 

NOO°-39'W 

NOO°-40V 

_*i  1=85-41' 

62 

34  -34 

61 

32  -34 

10560 

30  -34 

D  =  4°00' 

59 

28  -34 

1  =  71°22' 

58 

26-34 

57 
58 
55 

24-34 
22-34 
20-34 

CO 

8 
-f   5 

»o    g 
!o    TH 

0        II 

54 

0 

18-34 

PU 

53 

^ 

16-34 

52 

JA 

bo 

3 

14-34 

51 

p 

12-34 

10550 

** 

10-34 

43 

8  -34 

48 

6  -34 

47 

4-34 

46 

2-34 

45 

0  -34 

H-71.6 

O  T.C. 

0°-00' 

N72°-Ol'W 

N72°-10'W 

44 

43 

FIG.  5. 


J.E.Schmidt— Inst. 

R.Carr-H.C.  C.E.Agg— R.C. 

G.W.Pix— R.F. 

C.  Wiles  — Ax 


June  12, 13 
Clear-Warm 


C.T. 


)P.O.C. 


T.C. 


FIG.  5. 


31 


32  TEXT   BOOK   ON   RAILROAD    SURVEYING 

plates  at  the  total  deflection  angle  of  each  station  as  given  in  the 
notes,  and  measure  the  corresponding  distances. 

61.  The  FUNDAMENTAL  PRINCIPLES  are: 

When  sighting  at  any  station  the  plates  must  read  the  total  de- 
flection angle  of  that  station.  When  on  tangent  at  any  station  the 
plates  must  read  the  total  deflection  angle  of  that  station. 

If  these  principles  are  strictly  observed  a  mistake  in  instrument 
work  is  impossible. 

Example.  The  P.I.  of  two  tangents  is  at  Sta.  10555  +  00.3 
and  /  =  71°  22'.  It  is  desired  to  connect  the  tangents  with 
a  4°  curve. 

Tforal0  curve  for  /  =  71°  22' 


D 
4114.9  (from  Table  6) 


=  1028.7  ft. 


Sta.  T.C.  =  Sta.  P.I.  -  T  =  (10555  +  00.3)  -  (10  +  28.7)  = 

10544  +  71.6 

/        71°  22'       71.367 
L  =  -~-  =  — j —  =  — j—  =  17.842  Stations. 

Sta.  C.T.  =  Sta.  T.C.  +  L  =  (10544  +  71.6)  +  (17  +  84.2)  = 

10562  +  55.8 

The  distance  from  the  T.C.  to  the  first  Sta.  on  the  curve  is  28.4 
ft.,  hence 

di  =  0.3  CD  =  0.3  X  28.4  X  4  =  34' 
Deflection  angle  =  JD  =  2°  00' 

The  distance  from  the  last  Sta.  to  the  C.T.  is  55.8  ft.,  hence 
d2  =  0.3  X  55.8  X  4  =  67'  =  1°  07'  \ 

Total  Deflection  Angle  of         T.C.     =  0°  00' 

"      "  Sta.  10545  =  0°  00'  +  0°  34'  =  0°  34' 

"      ""      10546  =  0°  34'  +  2°  00'  =  2°  34' 

"  "  "      "  "      10547  =  2°  34'  +  2°  00'  =  4°  34' 


CURVES 


33 


Total  Deflection  Angle  of  Sta.  10562  =  =  34°  34' 

"  "  "      "         C.T.    =  34°  34'  +  1°07'  = 

35°  41'  =  i/  (check) 

These  results  are  recorded  in  the  form  shown  in  Fig.  5. 

62.  Offsets  from  chords  produced.  In  addition  to  his  chain- 
ing duties  the  head  chainman  also  sets  the  stakes  on  line  as 
directed  by  the  transitman.  Obviously  this  work  is  much 
facilitated  if  the  head  chainman  can  quickly  approximate  the 
position  of  the  stakes.  This  can  be  done  by  an  offset  from  the 
line  through  the  last  two  stations.  For  example,  in  Fig.  6,  A 
and  B  are  the  last  two  stations  set.  Produce  AB  to  E  the 
distance  C  from  B.  F  is  located  by  the  offset  y  from  E  and  the 
distance  C  from  B.  y  can  be  found  by  Eq.  (16).  If  E  is  care- 
fully lined  in  and  C  and  y  measured  with  the  tape,  the  point 
can  be  located  with  considerable  accuracy;  and,  if  a  transit 
is  not  at  hand,  this  method  can  be  employed  in  locating  an  entire 


FIG.  6. 


curve.  However,  this  method  is  most  valuable  as  an  aid  to  the 
head  chainman  in  approximating  the  position  of  the  stakes, 
and  it  should  always  be  used.  If  E  is  only  roughly  lined  in  and 
y  is  paced,  the  points  can  be  located  within  less  than  one  foot, 
which  is  sufficient  for  the  purpose. 
From  Fig.  6, 

\y  :  C:  :  J  C  :  R 


Whence 


\y  =     p  (C  and  R  expressed  in  feet) 


34  TEXT   BOOK   ON   RAILROAD   SURVEYING 

Expressing  R  in  terms  of  Z),  we  have 


*    ~  1)1460 
If  n  represents  C  expressed  in  stations,  then 

\y  =  0.873  n*D          (15) 

\y  is  the  tangent  offset  to  the  point  F. 

For  C  =  100  ft.,  n  =  1, 

and         \y  =  0.873  D,  and  y  =  1.746  D  =  if  D  (approx.)  (16,) 

For  C  =  50  ft.,  n  =  \, 

and  Jy  =  0.22  D,  and  y  =  0.44  D 

63.  Chord  Offset  Method.  There  are  some  cases  in  which  the 
deflection  angle  method  of  running  in  curves  can  not  be  used  to 
advantage.  For  instance,  in  heavily  wooded  country  it  may  not  be 
economical  to  locate  every  station  by  the  deflection  method  on 
account  of  the  large  amount  of  clearing  necessary  for  the  lines 
of  sight.  Fig.  7  illustrates  this  condition.  If  the  stakes  B,  C, 
E,  and  F  were  all  set  from  A  it  would  necessitate  the  clearing 


FIG.  7. 

of  a  line  to  each  or  practically  the  entire  area  between  the  curve 
and  the  chord  AF.  If,  on  the  other  hand,  F  were  set  from  A 
by  its  total  deflection  angle  and  the  chord  AF,  and  B,  C,  and  E 
located  by  offsets  from  this  chord,  the  only  clearing  required 
would  be  along  AF  and  the  short  offsets.  The  transit  would 
then  be  moved  to  F  and  the  process  repeated  to  J,  and  so  on. 

The  chord  AF  is  the  L.C.  of  a  curve  whose  value  of  /  is  the 
central  angle  of  the  curve  ABCEF,  and  therefore  can  be  de- 
termined from  Eq.  (3)  or  from  Table  6.  In  chaining  this  chord, 
temporary  stakes  are  set  at  the  full  station  distances,  although 
perpendicular  offsets  from  these  will  not  give  the  exact  location 


CURVES 


35 


of  the  curve  stations  since  the  chord  AF  is  shorter  than  the 
line  ABCEF.  This  error  may  be  neglected  for  the  following 
reasons:  (1)  the  error  at  any  point  in  practice  will  rarely  exceed 
one  foot;  (2)  the  perpendicu- 
larity of  the  offsets  is  estab- 
lished by  eye,  hence  is  not 
exact;  and  (3)  the  approxi- 
mate location  of  these  stations 
is  sufficient  since  this  method 
would  be  used  only  on  first 
location  to  determine  the 
profile.  The  offsets  are  com- 
puted as  follows: 

From  geometry  the  pro- 
ducts of  the  segments  of 
two  intersecting  chords  are  FlG  8 

equal.     Then  from  Fig.  8, 


or 


Si  XS2       S1X 


2R-S, 


Since  $3  is  small  compared  with  2R  it  may  be  neglected  in  the 
denominator,  and  since  in  practice  the  difference  between  $3  and 
the  offset  will  rarely -reach  0.05  ft.,  we  may  write 

Si  XS2       Si  XS.2XD  ,A 
Offset  =      nn —   =  —^^,r»^     (Approx.) 


2R 


2  X  5730 


Reducing  Si  and  $2  to  stations,  we  have 

Offset  (in  feet)  =  0.873  SiS2D     (Slide  Rule) 


(17) 


64.  Example.  The  T.C.  of  a  4°  curve  is  at  Sta.  67  +  82.7. 
Sta.  72  is  set  by  its  total  deflection  angle  (8°  21')  and  the  L.C. 
of  a  4°  curve  for  /  =  LD  =  4.173  X  4  =  16°  41.5',  which  from 
Table  6  =  415.8  ft.  While  chaining  this  chord,  temporary 
stakes  are  set  at  17.3,  117.3,  217.3,  and  317.3  ft.  from  the 
T.C.  From  Eq.  (17), 

The  offset  at  Sta.  68  =  0.873  X  0.173  X  4  X  4  =    2.4  ft. 

"      "    "    69  =  0.873  X  1.173  X  3  X  4  =  12.3  ft. 

"       "    "    70  =  0.873  X  2.173  X  2  X  4  =  15.2  ft. 

"      "    "    71  =  0.873  X  3.173  X  1  X  4  =  11.1  ft. 


36 


TEXT   BOOK   ON   RAILROAD   SURVEYING 


The  stations  on  the  curve  are  located  by  measuring  these  offsets 
from  the  temporary  stakes  already  set. 


Change  of  Location  Problems 

After  the  line  is  located  according  to  the  paper  projection  and 
a  profile  made,  it  may  be  found  that,  owing  to  inaccuracies  in  the 
platted  topography,  a  considerable  cut  or  fill  comes  on  a  hillside 
and  may  be  eliminated  by  shifting  a  portion  of  the  alinement. 
Occasionally  other  considerations  than  earthwork  make  euch  a 
shift  desirable. 

65.  Problem  1.  In  Fig.  9,  the  line  as  originally  located 
(shown  dotted)  involves  considerable  earthwork  which  could 


FIG.  9. 

be  eliminated  by  shifting  the  curve  the  distance  p  at  or  near  its 
middle  point. 

The  problem  is  to  find  the  new  degree  of  curve  and  the  change 
in  the  position  of  the  T.C.  and  the  C.  T.  p  is  the  difference  in  the 
external  distances  of  the  two  curves,  or 

f^new  curve   ~   -^old  curve     i     P 

The  new  degree  of  curve  may  be  computed  from  the  new 
external  by  Eq.  (5)  or  by  means  of  Table  6,  which  is  the  quicker 


CURVES 


37 


and  more  common  method.  The  estimated  value  of  p  will 
usually  give  an  odd  value  of  D  which  is  undesirable;  hence  the 
value  of  D  to  be  used  is  taken  only  to  the  nearest  ten  minutes. 
The  distance  that  the  T.C.  and  the  C.T.  are  shifted  is  equal  to 
the  difference  in  T  for  the  two  curves. 

66.  Problem  2,  Case  1. 

In  Fig.  10,  the  line  as  originally  located  (shown  dotted)  in- 
volved considerable  earthwork  between  A  and  B,  which  could 


FIG.  10. 


largely  be  eliminated  by  shifting  the  tangent  the  distance  p 
parallel  to  its  original  position,  which  involves  a  change  in  the 
location  of  TWO  curves. 

The  problem  is  to  find  the  change  in  the  T.C.s  and  the  C.T.s 
of  the  two  curves,  the  degree  of  curve  remaining  unchanged.  In 
triangle  1,  the  sides  m  and  I  can  be  readily  computed  since  / 
and  p  are  known,  m  =  n  =  the  required  change  in  the  T.C. 
and  P.I.  p  will  rarely  exceed  100  feet,  hence  the  new  tangent 
is  best  located  by  offsets  from  the  old  tangent.  The  new  C.T. 
is  then  located  from  the  old  C.T.  by  rectangular  coordinates 
one  of  which  is  p  and  the  other  is  the  side  I  of  triangle  1. 

The  second  curve  is  located  similarly  using  triangle  2. 


38 


TEXT   BOOK    ON   RAILROAD    SURVEYING 


67.  Problem  2,  Case  2. 

Owing  to  some  special  cause,  for  example  a  stream-crossing 
as  shown  in  Fig.  11,  it  may  be  undesirable  to  move  the  C.T.  for- 
ward as  in  Case  1.  This  condition  can  usually  be  met  by  placing 


FIG.  11. 


the  new  C.T.  approximately  on  the  same  radial  line  as  the  old 
C.T.  This  involves  a  change  in  the  degree  of  curve  and  in  the 
position  of  the  T.C. 

In  triangle  1,  7  and  p  are  known,  and  I  and  m  are  computed. 

7  new  curve  =   1  old  curve          ' 

whence  the  new  D  is  easily  found  from  Table  6  and,  as  in  Problem 
1,  is  taken  only  to  the  nearest  10  minutes.  (The  approximation 
in  the  position  of  the  C.T.  is  due  to  thus  not  using  the  exact 
value  of  D.)  A  new  value  of  T  is  then  found  which  agrees  with 
the  D  that  is  used.  Then 

^  =   -*  old  curve  ~{~  Wl  —  2  new  curve 

68.  Problem  2,  Case  3.  This  case  is  similar  to  Case  2, 
except  that  the  C.T.  is  on  the  fixed  tangent  instead  of  the  one 
being  changed,  hence  the  solution  is  somewhat  different. 

Solve  triangle  2  for  the  side  m,  then 

-*  new  curve  =    Told  curve    —  W 

whence  the  new  degree  of  curve  is  obtained  from  Table  6. 


CURVES  39 

Solve  triangle  3  for  the  side  1.  Then  the  new  T.C.  can  be 
located  from  the  old  T.C.  by  the  co-ordinates  I  and  p. 

As  in  the  previous  problem,  D  should  be  taken  only  to  the 
nearest  ten  minutes.  This  necessitates  a  slight  change  in  both 
the  T.C.  and  the  C.T.,  since  T  for  this  value  of  D  is  not  the 
same  as  that  used  in  the  abpve  computations.  It  must  be 
remembered  that  the  T.C.  becomes  the  C.77.,  or  vice  versa,  when 
the  line  is  run  in  the  opposite  direction. 

69.  Problem  3.  Sometimes  curves  are  run  in  beginning  at 
the  T.C.  without  previously  locating  the  P. I.  and  the  C.T.,  and 
checking  the  direction  of  the  forward  tangent.  On  reaching 
the  C.T.  and  projecting  the  forward  tangent,  it  is  found  that 
the  tangent  does  not  pass  through  some  controlling  point. 
It  is  therefore  necessary  to  change  the  direction  of  the  tangent 
by  shifting  the  C.T.  This  is  most  readily  accomplished  by 
measuring  the  necessary  change  of  direction  with  the  transit, 
and  then  changing  the  length  of  the  curve  to  correspond.  For 
example,  in  Fig.  12,  the  curve  is  run  to  B  as  a  C.T.  and  the 
tangent  is  found  to  pass  through  x,  whereas  it  is  necessary  to 
pass  through  A.  The  angle  e  is  measured  with  the  transit. 


The  length  of  curve  corresponding  to  e  is  BC,  whence  the  C.T. 
is  moved  back  to  C.  The  forward  tangent  then  takes  the 
position  CA',  parallel  to  BA,  and  misses  A  by  the  distance  m, 
which  is  the  middle  ordinate  of  the  curve  BE. 


40  TEXT  BOOK   ON   RAILROAD   SURVEYING 

BE  =  2BC  = 


m  =  -g-X  X  D  =  0.875-- 

m  will  rarely  reach  5  ft.,  hence  the  line  CA'  meets  the  require- 
ments. 

Problems 

1.  If  the  P.I.  is  in  a  stream  or  is  otherwise  inaccessible,  how 
can  /  and  the  positions  of  the  T.C.  and  the  C.T.  be  determined? 

2.  If  the  T.C.  is  inaccessible,  how  can  the  curve  be  run  in  and 
checked? 

3.  If  the  line  of  a  curve  passed  through  a  building  or  similar 
obstruction,  how  can  the  curve  be  run  in  and  checked? 

4.  Given  I  =  63°  43'.     Find  T,  L,  R,  and  E  for  each  of  the 
following  values  of  D.—  0°  30',  2°  00',  5°  00',  and  7°  30'.     Solve 
by  equation  and  check  by  Table  6. 

5.  Given  the  P.I.  at  Sta.  118  +  60.0,  /  =  57°  48',  and  D  = 
5°  00'.     Find  the  station  numbers  of  the  T.C.  and  the  C.T. 

6.  Write  transit  notes  for  the  preceding  problem. 

7.  Given,  on  original  location,  a  6°  00'  curve  with  /  =  78°  21' 
and  the  T.C.  at  Sta.  1041   +  72.6.     Topography  requires  the 
middle  point  of  the  curve  to  be  moved  toward  the  center  about 
45  ft.     What  is  the  new  degree  of  curve  and  the  station  number 
of  its  T.C.? 

8.  Given  an  established  curve  with  7  =  69°  38',  D  =  3°  00', 
and  the  T.C.  at  Sta.  982  +  41.1.     It  is  desired  to  move  the 
forward  tangent  outward  a  distance  of  50   ft.     Find  the  new 
station  numbers  of  the  T.C.  and  the  C.T7.,  and  the  coordinates 
from  the  old  C.T.  to  the  new  C.T. 

Answer.     T.C.  =  982  +  94.4;  C.T.  =  1006  +  15.5;  I  =  18.6, 
and  p  =  50.0. 

9.  Data  same  as  in  Problem  8,  except  that  the  forward  tangent 
is  moved  inward. 

Answer.  T.C.  =  981  +  87.8;  C.T.  =  1005  +  08.9;  I  =  18.6, 
and  p  =  50.0. 

10.  Given  an  established  curve  with  /  =  47°  23',  D  =  2°  30', 
and  the  T.C.  at  Sta.  1841  +  83.7.     It  is  desired  to  move  the 


CURVES  41 

forward  tangent  outward  a  distance  of  60  ft.,  keeping  the  C.T. 
on  the  same  radial  line  approximately.  Find  the  new  degree  of 
curve,  the  station  number  of  the  T.C.,  and  the  distance  that 
the  C.T.  moves  forward  or  backward  due  to  using  D  to  the  near- 
est ten  minutes  only. 

Answer.  D,  computed  =  2°  38.7';  Z),  used  =  2°  40';  T.C.  ~ 
1843  +  28.0;  C.T.  moves  7.6  ft.  backward. 

11.  Data  same  as  in  Problem  10,  except  that  the  forward 
tangent  is  moved  inward. 

Answer.  D,  computed  =  2°  22.2';  D,  used  =  2°  20';  T.C.  = 
1840  +  30.3;  C.T.  moves  16.7  ft.  forward. 

12.  Given  an  established  curve  with  /  =  50°  35',  D  =  4°  30', 
and  the  T.C.  at  Sta.  155  +  24.5.     It  is  desired  to  move  the 
initial  tangent  35  ft.  inward  without  changing  the  position  of  the 
C.T.  (except  the  small  amount  due  to  using  approx.  value  of  D). 
Find  the  new  degree  of  curve,  the  station  numbers  of  the  new 
T.C.  and  C.T.,  the  coordinates  of  the  new  T.C.  from  the  old 
T.C.,  and  the  small  amount  that  the  T.C.  and  the  C.T.  must 
shift  due  to  the  approx.  value  of  D  used. 

Answer. .D,  computed  =  4°  52';  D,  used  =  4°  50';  T.C., 
computed  =  155  +  98.7;  T.C.,  used  =  155  +  94.9;  C.T7.,  com- 
puted =  166  +  38.1;  C.T.,  used  =  166  +  41.5;  I  =  70.3;  p  = 
35.0;  T.C.  and  C.T.  shifted  3.8  ft. 

13.  Data  same  as  in  Problem  12  except  that  the  tangent  is 
moved  outward. 

Answer.  D,  computed  =  4°  11';  D,  used  =  4°  10';  T.C., 
computed  =  154  +  50.3;  T.C.,  used  =  154  +  47.5;  C.T.,  com- 
puted =  166  +  59.2;  C.T.,  used  =  166  +  61.5;  I  =  77.0;  p  = 
35.0;  T.C.  and  C.T.  shifted  2.8  ft. 

14.  After  running  in  a  3°  00'  curve  to  the  right,  using  paper 
location  notes,  it  is  found  that  the  forward  tangent  passes  to  the 
right  of  the  governing  point  by  2°  20'  as  measured  at  the  C.T. 
(a)  What  distance  must  the  C.T.  be  shifted?     (b)  How  much 
does  the  new  tangent  miss  the  controlling  point? 

Answer,     (a)  =  77.8ft.;  (b)  =  1.6ft. 

Compound  Curves 

70.  A  compound  curve  is  a  combination  of  two  or  more  simple 
curves  in  the  same  direction  with  a  common  tangent  at  the 


42 


TEXT  BOOK   ON    iAILROAD   SURVEYING 


point  of  junction.  A  compound  curve  should  never  be  used  except 
under  conditions  where  a  simple  curve  will  not  meet  the  require- 
ments. In  rough  country  it  may  happen  that  a  large  volume  of 
earthwork  can  be  avoided  by  making  one  part  of  a  curve  sharper 
than  another,  resulting  in  a  compound  curve. 

For  example,  in  Fig.  13,  the  line  is  located  as  a  compound  curve 
ABC,  requiring  little  earthwork  and  keeping  on  the  bank  of 
the  stream.  If  the  simple  curve  AB  had  been  produced  to  F  to 


end  in  a  parallel  tangent,  it  would  have  fallen  in  the  river.  Or 
if  the  simple  curve  CB  had  been  prolonged  to  E,  it  would  have 
pierced  the  cliff. 

The  degree  of  curve  and  the  point  of  compound  curve,  C.C., 
are  chosen  to  fit  the  contours  and  other  governing  conditions, 
and  the  central  angles  of  the  branches  are  scaled  from  the  map. 

This  is  one  of  many  cases  where  a  compound  curve  is  ap- 
plicable. 

From  the  standpoint  of  operation  a  compound  curve  is  better 
than  two  simple  curves  separated  by  a  short  tangent,  hence  it 
should  be  used  in  such  cases.  In  flat  country  a  compound  curve 
is  inexcusable  on  main  line.  If  the  degrees  of  curve  of  two  ad- 
jacent branches  of  a  compound  curve  differ  by  more  than 
2°  00',  an  easement  curve  should  be  inserted  between  the  branches, 
and  provision  for  this  should  be  made  in  locating  the  curve. 
(See  Spirals,  paragraph  84.) 


CURVES 


43 


The  nomenclature  and  positions  of  the  functions  are  shown 
in  Fig.  14. 

71.  Problem  1. 

In  new  work  /  will  be  measured  and  /i,  72,  A,  and  A  will  be 
determined  from  the  preliminary  maps.     It  is  desired  to  know 


the  value  of  T\  and  T2  in  order  to  set  the  T.C.  and  the  C.T.  and 
to  run  in  the  curve. 

In  Fig.  14,  BE  is  the  common  tangent  at  the  C.C. 

BE  =  BC  +  CE 

BC  =  AB  is  the  tangent  distance  for  a  A  curve  for  an  inter- 
section angle  I\\  and  CE  =  #G  is  the  tangent  distance  for  a  A 
curve  for  an  intersection  angle  72  and  are  computed  by  Table  6. 
Then  in  the  triangle  BEF,  one  side  and  the  adjacent  angles  are 
known  from  which  the  sides  BF  and  EF  are  computed. 

Tl  =  AB  +  BF,  and  T2  =  EF  +  #£ 

72.  Problem  2.  Occasionally  the  positions  of  the  T.C.  and 
the  C.T.  are  fixed  within  narrow  limits,  giving  unequal  tangent 
distances,  which  necessitates  a  compound  curve. 


44 


TEXT   BOOK    ON   EAILROAD    SURVEYING 


The  degree  of  curve  of  one  branch  must  be  chosen  before  the 
problem  can  be  solved.     The  problem  then  is  to  determine  the 


FIG.  15. 


degree  of  curve  of  the  second  branch  and  the  central  angles 
/i  and  72. 

In  Fig.  15,  /,  Ti,  Ti,  and  A  are  known.     A  H  =  HF  =  T0  = 
the  tangent  of  A  curve  for  the  intersection  angle  /. 


In  triangle  1, 


m  =  To  -  Tl 


and  hence  the  triangle  can  be  solved  for  p  and  n. 
In  triangle  2, 

I  =  T0  +  n  -  T2 

Since  p  is  known,  the  triangle  can  be  solved  for  the  angle  at 
E,  which  is  i/2. 

/i  =  /  -  h 

In  triangle  3,  since  I  and  72  are  known,  the  side  Oi02  =  Ri  —  Rz 
can  be  solved  for,  whence  Z>2  is  determined  from  Table  5. 


CURVES 


45 


73.  Field  Work. 

A  point  on  one  branch  of  a  compound  curve  can  not  be  located 
with  the  transit  set  on  a  point  on  another  branch,  since  in  order 
to  run  in  a  circular  curve  with  the  transit  and  tape,  the  transit 
and  the  points  to  be  located  must  be  on  the  same  circumference. 
Therefore  each  branch  of  a  compound  curve  is  run  in  indepen- 
dently as  a  simple  curve.  Thus  the  notes  for  the  first  branch  are 
computed  and  the  curve  run  in  to  the  C.C.  The  transit  is  then 
placed  at  the  C.C.  and  oriented  as  at  the  C.T.  of  a  simple  curve 
except  that  the  plates  read  zero  instead  of  ^1  when  sighting  along 
the  tangent*  The  notes  for  the  second  branch  are  then  com- 
puted and  the  curve  run  in  to  the  C.  T.  or  to  the  next  C.C. 

Change  of  Location  Problems 

74.  Problem  1.     This  problem  arises  under  conditions  similar 
to  those  in  Problem  2,  Case  1,  of  simple  curves,  paragraph  66. 

Given,  as  shown  in  Fig.  16,  a  located  compound  curve  ABC. 
It  is  desired  to  move  the  forward  tangent  laterally  a  distance  p. 


FIG.  16. 

The  simplest  solution  of  this  problem  is  to  retain  the  same 
degrees  of  curve  and  to  shift  the  C.C.  from  B  to  E. 

BE  (in  feet)  =  -^T  1Q0 
L>i 

*  To  orient  at  the  C.C.,  back-sight  on  the  last  transit  station  with  the 
plates  set  to  read  the  difference  between  the  total  deflection  angle 
of  the  C.C.,%  Ji,  and  the  total  deflection  angle  of  the  back-sight  sta- 
tion; turn  the  plates  to  zero  and  the  line  of  sight  is  on  tangent. 


46 


TEXT   BOOK    ON   RAILROAD    SURVEYING 


Therefore  the  problem  resolves  itself  into  solving  for  the  angle  b. 

In  triangle  1,  since  a  is  known  and  the  hypotenuse  is  the 
difference  in  the  two  radii,  the  base  m  can  be  computed.  Then 
n  =  m  —  p. 

In  triangle  2,  the  hypotenuse  and  the  base  are  known,  whence 
the  angle  c  is  computed. 

b  =  c  —  a 

In  triangle  3,  e  =  90°  —  J6  —  a  (prove) 

whence  the  coordinates  of  M,  the  new  C.T.,  from  the  old  C.T. 
can  be  determined. 

Three  other  cases  of  this  problem  arise,  viz.:  (1)  the  same  as 
above,  except  that  the  tangent  is  moved  outward;  (2)  the  curve  of 
longer  radius  may  fall  on  the  tangent  to  be  moved  inward;  and 
(3)  the  curve  of  longer  radius  may  fall  on  the  tangent  to  be 
moved  outward. 

The  solution  of  each  of  these  cases  is  the  same  as  that  given 
above  except  that  the  various  points  take  different  relative 
positions.  If  the  same  relative  construction  lines  are  drawn  and 
the  corresponding  points  are  used,  no  confusion  should  arise. 

75.  Problem  2.  This  problem  arises  under  conditions  similar 
to  those  in  Problem  2,  Case  2,  of  simple  curves,  paragraph  67. 


\ 


FIG.  17. 


Given,  as  shown  in  Fig.  17,  a  located  compound  curve  ABC. 
It  is  desired  to  move  the  forward  tangent  laterally  a  distance  p, 


CURVES  47 

and  to  keep  the  C.T.  approximately  on  the  same  radial  line, 
which  involves  a  change  in  the  degree  of  curve  of  the  second 
branch  and  in  the  position  of  the  C.C. 

In  triangle  1,  a  is  known  and  the  hypotenuse  is  the  difference 
in  the  two  given  radii.  Therefore  the  base  m  and  the  altitude 
n  can  be  computed. 

In  triangle  2,  the  base  n  is  now  known  and  the  altitude  I  can  be 
determined  since  R\y  R^  p,  and  m  are  known.  The  angle  at 
p  =  |6  (Why?)  can  therefore  be  determined.  Then 

c  =  b  —  a 

and  the  C.C.  is  shifted  the  distance  BE  (in  feet)  =  -£- 100. 

DI 

In  triangle  3,  the  angle  b  and  the  altitude  n  being  known,  the 
hypotenuse  r  can  be  computed  from  which  Rx  is  determined,  and 
the  corresponding  degree  of  curve  Dx  is  found  from  Table  5, 
and  is  taken  to  the  nearest  ten  minutes.  Then  since  E  and  b 
are  already  fixed,  this  will  change  the  value  of  p  slightly  and  will 
shift  the  C.T.  a  short  distance  along  the  line  EG  from  F,  its 
theoretical  location.  The  coordinates  of  this  new  position  are 
given  by  the  equations 

x  =  h  sin  b,     and     y '  —  x  tan  J6, 

in  which  x  is  the  movement  along  the  tangent,  y  is  the  change  in 
p,  and  h  is  the  difference  between  the  computed  value  of  Rx  and 
the  value  actually  used. 

Three  other  cases  of  this  problem  arise,  depending  on  whether 
the  longer  or  the  shorter  radius  curve  is  on -the  tangent  to  be 
moved,  and  on  whether  the  tangent  moves  inward  or  outward. 
In  all  of  these  cases  the  solution  is  relatively  the  same. 

Problems 

1.  Given  the  P. I.  at  Sta.  837  +  00,  7  =  64°  44',  A  =  29°  00', 
72  =  35°  44',  A  =  4°  00',  and  D2  =  5°  30'.     Find  the  station 
numbers  of  the  T.C.,  C.C.,  and  C.T. 

Answer.  T.C.  =  828  +  73.4;  C.C.  =  835  +  98.4;  C.T.  =  842 
+  73.4. 

2.  Given  the  P.I.  at  Sta.  1846  +  50.0,  /  =  57°  18',  Ti  =  835.0, 


48  TEXT   BOOK    ON    RAILROAD    SURVEYING 

T2  =  687.0,  and  A  =  3°  00'.     Find  /i,  72,  A,  and  the  station 
numbers  of  the  T.C.,  C.C.,  and  C.T. 

Answer,     h  =  16°  17';  72  =  41°  01';  Z)2  =  4°  47.5';  T.C.  = 
1838  +  15.0;  C.C.  =  1843  +  57.8;  C.T.  =  1852  +  13.5. 

3.  A  2°  30'  curve  compounds  with  a  4°  00'  curve  at  Sta.  8792 
-f  27.6.     The  central  angle  of  the  4°  00'  curve  =  26C  45'.     It 
is  desired  to  move  the  forward  tangent  inward  30  ft.,  but  to  re- 
tain the  same  degree  of  curve.     Find  the  station  number  of  the 
new  C.C.  and  the  coordinates  of  the  new  C.T.  referred  to  the 
old  C.T. 

Answer.     C.C.  =  8790  +  61.6;  I  =  54.5  ft.;  p  =  30.0  ft. 

4.  Data  the  same  as  in  Problem  3,  except  that  the  tangent  is 
to  be  moved  outward. 

Answer.     C.C.  =  8794  +  22.0;  I  =  66.4  ft.;  p  =  30.0  ft. 

5.  A  5°  00'  curve  compounds  with  a  3°  00'  curve  at  Sta.  147 
+  63.3.     The  central  angle  of  the  3°  00'  curve  is  19°  00'.     The 
forward  tangent  is   to  be  moved  25  ft.  inward,  but   the   same 
degree  of  curve  is  to  be  retained.     Find  the  station  number  of 
the  new  C.C.  and  the  coordinates  of  the  new  C.T.  referred  to 
the  old  C.T. 

Answer.     C.C.  =  149  +  04.0;  I  =  90.2  ft.;  p  =  25.0  ft. 

6.  Data  same  as  in  Problem  5,  except  that  the  tangent  is  to  be 
moved  outward. 

Answer.     C.C.   =  146  +  61.3;  I  =  63.3ft.;  p  =  25.0ft. 

7.  A  4°  30'  curve  compounds  with  a  7°  30'  curve  at  Sta. 
999  +  67.0.     The  central  angle  of  the  7°  30'  curve  is  39°  24'. 
The  forward  tangent  is  to  be  moved  inward  50  ft.  and  the  C.  T. 
is  to  be  kept  approximately  on  the  same  radial  line.     Find  the 
new  degree  of  curve  for  the  second  branch,  the  station  number 
of  the  new  C.C.,  and  the  coordinates  of  the  new  C.T.  (actual) 
referred  to  the  old  C.T. 

Answer.     D,  computed  =  6°  33';  D,  used  =  6°  30';  C.C.  = 
996  +  36.2;  x  =  5.4  ft.  forward;  p'  =  47.2  ft. 

8.  Data  same  as  in  Problem  7,  except  that  the  tangent  is  to  be 
moved  outward. 

Answer.     D,   computed    =    12°  51.3';   D,  used    =    12°  50'; 
C.C.  =  1003  +  31.4;  x  =  0.3  ft.  forward;  P'  =  50.0. 

9.  A  5°  30'  curve  compounds  with  a  3°  40'  curve  at  Sta.  1888 
+  36.2.     The  central  angle  of  the  3°  40'  curve  is  27°  50'.     The 


CURVES  49 

forward  tangent  is  to  be  moved  inward  30  ft.  and  the  C.T. 
is  to  be  kept  approximately  on  the  same  radial  line.  Find  the 
new  degree  of  curve  for  the  second  branch,  the  station  number 
of  the  new  C.C.,  and  the  coordinates  of  the  new  C.T.  (actual) 
referred  to  the  old  C.T. 

Answer.  D,  computed  =  2°  49.0r;  D,  used  =  2°  50';  C.C.  = 
1890  +  84.4;  x  =  3  ft.  backward;  p'  =  30.4  ft. 

10.  Data  the  same  as  in  Problem  9,  except  that  the  tangent 
is  to  be  moved  outward. 

Answer.  D,  computed  =  4°  03.0';  D,  used  =  4°  00';  C.C.  = 
1886  +  01.5;  x  =  11.7  ft.  forward;  pf  =  34.3  ft. 

Reversed  Curves 

76.  A  reversed  curve  is  a  combination  of  two  simple  curves 
of  opposite  curvature  with  a  common  tangent  at  the  point  of 
junction. 

A  reversed  curve  should  never  be  used  on  main  lines  on  account 
of  the  shock  due  to  the  sudden  reversal  of  curvature  and  also  due 
to  the  fact  that  it  is  impossible  to  superelevate  properly  the 
outer  rail  at  or  near  the  point  of  reversal.  If  conditions  require 
two  curves  of  contrary  curvature  close  together,  they  should  be 
separated  by  sufficient  tangent  to  run  out  the  superelevation  of 
each,  or  it  should  be  provided  with  easement  curves  in  which 
case  the  "  points  of  spiral"  may  be  coincident. 

In  yards  and  connections,  reversed  curves  may  be  permissible, 
since  the  speed  is  low,  and  in  some  cases  may  be  imperative. 
In  nearly  every  case,  however,  they  should  be  employed  in  con- 
junction with  turnouts,  therefore  further  discussion  of  reversed 
curves  will  be  included  with  the  problems  of  turnouts  and  con- 
nections in  Chapter  VI. 

Part  2.     Spirals 

77.  A  spiral  is  a  curve  of  varying  radius  which  is  used  at  the 
ends  of  circular  curves  in  order  that  the  change  from  rectilinear 
to  circular  motion,  or  vice  versa,  may  be  gradual  and  without 
shock.     It  is  also  used  between  the  branches  of  a  compound 
curve  for  similar  reasons. 

On  tangents  the  track  should  be  level  transversely,  but  on 
curves  the  outer  rail  is  elevated  above  the  inner  one  to  counter- 


50 


TEXT   BOOK   ON   RAILROAD   SURVEYING 


ict  the  effect  of  centrifugal  force.  If  this  were  carried  out  lit- 
erally, there  would  be  a  vertical  jog  in  the  outer  rail  at  the  T.C. 
and  the  C.T.,  which  is  impossible  from  the  standpoint  of  opera- 
tion. It  is  evident  then  that  the  superelevation  must  be  at- 
tained gradually;  the  distance  in  which  this  is  done  is  called 
the  "  run-off."  It  is  the  custom  on  unspiraled  curves  to  have 
the  run-off  either  wholly  or  in  part  on  the  tangent  with  the 
result  that  there  is  a  disagreeable  tipping  of  the  train  upon 
entering  and  leaving  curves.  When  a  spiral  is  used  this  objec- 
tion is  overcome  since  the  superelevation  is  attained  in  its  length, 
and  at  every  point  is  correct  for  the  degree  of  curve  at  that  point. 
Thus  the  spiral  may  be  defined  as  a  curve  whose  degree  of  curve 
increases  uniformly  from  zero  at  Us  beginning  to  the  degree  of  the 
main  curve  at  its  end. 


Application  to  Simple  Curves 

78.  The  functions  of  the  spiral  are  shown  in  Fig.  18,  and  are 
defined  as  follows: 

T.S. — the  point  of  change  from  tangent  to  spiral. 
S.C. — the  point  of  change  from  spiral  to  circular  curve. 


FIG.  18. 


C.S. — the  point  of  change  from  circular  curve  to  spiral. 
S.T. — the  point  of  change  from  spiral  to  tangent. 

a  — the  angle  between  the  tangent  at  the  T.S.  and  a  chord  to 
any  point  on  the  spiral — the  spiral  deflection  angle. 


CURVES  51 

A  —  a  for  the  S.C.  —  the  spiral  total  deflection  angle. 

b  —  the  angle  at  any  point  on   the  spiral  between   a  tangent 

at  that  point  and  a  chord  to  the  T.S. 
B—b  at  the  S.C. 

d  —  degree  of  curve  of  the  spiral  at  any  point. 
D  —  degree  of  curve  of  the  central  circular  curve. 
/  —  the  angle  at  any  point  on  the  spiral  between  a  tangent 

at  that  point  and   a  chord  to  any  other   point  (b  is   a 

special  case  of  /). 

7  —  the  total  central  angle  of  the  spiraled  curve. 

d  —  the  central  angle  of  the  spiral  from  the  T.S.  to  any  point. 
A  —  the  central  angle  of  the  spiral  from  the  T.S.  to  the  S.C. 
k  —  the  rate  of  change  in  the  degree  of  curve  of  the  spiral  per 

station. 

I  —  the  length  of  the  spiral  in  feet  from  the  T.S.  to  any  point. 

L  —  the  length  of  the  spiral  in  feet  from  the  T.S.  to  the  S.C. 

s  —  the  length  of  the  spiral  in  stations  from  the  T.S.  to  any 

point  =  JQJJ. 

8  —  the  length  of  the  spiral  in  stations  from  the  T.S.  to  the 


r  —  the  radius  of  curvature  of  the  spiral  at  any  point  (radius 

corresponding  to  d). 

R  —  the  radius  of  curvature  of  the  central  circular  curve. 
x  —  the  abscissa  of  any  point  referred  to  the  T.S. 
X—  the  abscissa  of  the  'S.C.  referred  to  the  T.S. 
y  —  the  ordinate,  or  tangent  offset,  of  any  point. 
Y  —  the  ordinate,  or  tangent  offset,  of  the  S.C. 
T.C.  —  the  point  of  curve  of  central  curve  produced  back  to  a 

tangent  parallel  to  the  tangent  at  the  T.S. 
o  —  the  ordinate  of  the  T.C. 
t—  the  abscissa  of  the  T.C.  referred  to  the  T.S. 
P.I.  —  the  point  of  intersection  of  the  tangents  of  the  spiraled 

curve. 
Ts  —  the  tangent  distance  of  the  spiraled  curve  (T.S.  to  P./.). 


52  TEXT   BOOK    ON   RAILROAD    SURVEYING 

T  —  the  tangent  distance  of  an  unspiraled  curve  of  the  same 

D  and  7. 

Es  —  the  external  distance  of  the  spiraled  curve. 
E  —  the  external  distance  of  the  unspiraled  curve  of  the  same 
D  and  /. 

79.  Formulas. 

From  definition, 

kl 

d  =  ks  =  loo 


The  derivations  of  the  following  formulas  require  the  use  of  the 
calculus  and  are  given  in  the  Appendix. 

d  (in  degrees)    =  Jks2,  A  =  |kS2  .......     (19) 

a   (in  degrees)    =  J  d  =  |  ks2        ......       ) 

a   (in  minutes)  =  lOks2,  >•    (20) 

A  (in  degrees)  =  f  A  =  J  kS2  ) 

b  (in  degrees)    =  f  d  =  2a, 

B  (in  degrees)  =  f  A  =  2  A    ' 
y    =  0.291  ks3    -  .00000158kV      [  (     ^ 

Y  =  0.291  kS3  -  .00000158  k3S7  f 

x    =  1    -  .000762  k2s    )  /x 

X  =  L  -  .000762  k2S5  f 
o    =  0.0727  kS3  ..........      .      .     (24) 

t    =  iL  -  .000127  k2S5       .........      (25) 

Ts=  T  +  otan  JI  +  t  =  (R  +o)  tanfl  +  t    .      .      .     (26) 

ES  =  E  +  ^~Tf  =  (R  +  o)  exsec  JI  +  o      .      .      .      .     (27) 

COS   2-*- 

80.  Length  of  Spiral. 

Since  the  function  of  a  spiral  is  to  ease  the  entrance  to  a  cir- 
cular curve,  D  will  always  be  known.  Then  from  Eq.  18  it  is 
only  necessary  to  choose  a  value  of  k  or  L,  and  the  spiral  is  fixed. 
Obviously  any  value  of  k  could  be  chosen  and  the  spiral  would 
fit,  but  such  a  spiral  may  be  so  short  as  to  require  an  excessive 


CURVES  53 

rate  of  superelevation,  or,  on  the  other  hand,  it  may  be  needlessly 
long.  It  is  therefore  more  logical  to  determine  the  length  of  spiral 
necessary  to  give  the  desired  rate  of  superelevation  and  to  make 
k  to  correspond. 

The  length  of  the  spiral  is  a  direct  function  of  the  rate  of 
superelevation.  The  total  superelevation  depends  upon  the 
speed  and  the  degree  of  curve  and  is  found  from  the  formula, 

*e  =  0.00069  DV2 (28) 

in  which  e  is  the  superelevation  in  inches  and  V  the  velocity  in 
miles  per  hour,  e  should  never  exceed  8  inches  on  account  of  the 
effect  on  slowly  moving  trains,  but  the  track  should  be  super- 
elevated  for  the  fastest  tram  up  to  this  limit.  Therefore  the 
maximum  train  speed  for  any  given  curve  is  limited  by  this 
maximum  superelevation.  According  to  the  American  Railway 
Engineering  Association,  curves  requiring  less  than  2  inches  of 
superelevation  are  not  generally  spiraled,  as  the  run-off  may  be 
placed  on  the  tangent  without  objection. 

The  maximum  rate  at  which  the  superelevation  may  be  at- 
tained without  discomfort  to  passengers  is  about  1  1-6  inches 
per  second. f  Therefore  the  minimum  length  of  spiral  is, 

L  =  6/7  ev 
where  v  is  in  feet  per  second.     Reducing  v  to  miles,  per  hour, 

L  =  1.26  eV 

Substituting  the  value  of  e  from  Eq.  28, 

L  =  . 00087  DV8      ....     (29) 

Substituting  the  value  of  D  from  Eq.  18,  and  solving  for  k,  we 
have 

115000.,  ,QrA 

k  =      V3      (Approx.)    ....     (30) 

The  diagram,  Fig.  19,  is  platted  from  values  computed  by  the 
above  formulas,  and  gives  the  length,  superelevation,  and  k  for 

*  For  derivation,  see  Appendix,  paragraph  143. 

t  American  Railway  Engineering  Association,  Bulletin  108,  Feb., 
1911. 


TEXT   BOOK    ON   RAILROAD    SURVEYING 


CURVES 


55 


curves  from  0°  30'  to  10°  00',  and  for  speeds  from  20  to  80  miles 
per  hour.  In  this  diagram,  with  any  given  value  of  D  and  F, 
the  length  of  spiral,  the  corresponding  value  of  ft,  and  the  re- 
quired superelevation  can  be  readily  determined. 

The  consensus  of  opinion  of  31  railroads  of  the  United  States  as 
reported  to  the  American  Railway  Engineering  Association  is  that 
the  theoretical  speed  for  which  the  outer  rail  is  superelevated,  or 
the  curve  spiraled,  may  be  exceeded  about  20  per  cent,  without 
discomfort  to  passengers.  Further,  the  maximum  speed  is 
never  accurately  known.  Therefore,  it  is  not  necessary  to  use 
the  exact  values  of  k  and  L  as  determined  for  the  assumed  speed. 
Obviously,  even  values  of  k  are  desirable,  and  since  considerable 
variation  is  permissible,  such  values  may  be  chosen,  provided 
the  change  does  not  increase  the  speed  more  than  20  per  cent. 
An  examination  of  the  diagram,  Fig.  19,  will  show  that  values  of 
A;  of  i,  1,  2,  and  4  will  give  a  satisfactory  length  of  spiral  for 
practically  all  cases  met  in  practice. 

81.  Field  Work. 

The  first  step  in  the  field  work  is  to  determine  the  station 
numbers  of  the  T.S.,  S.C.,  C.S.,  and  S.T.  The  T.S.  and  the 


FIG.  20. 

S.T.   are   located   on   the   ground   by   measuring  the   distance 
!Fs  from  the  P.I. 
From  Fig.  20, 

Ts  =  T  +  o  tan  ^1  +  t 


(31) 


56  TEXT   BOOK   ON   RAILROAD   SURVEYING 

T  is  found  from  Table  6  (or  Eq.  2);  o  is  found  from  Table  1 
(or  Eq.  24);  and  t  is  found  from  Table  1  (or  Eq.  25). 

Example.  Given:— P.I.  =  21  +  21.1,  D  =  3°  40',  /  = 
51°  20',  and  maximum  speed  about  45  miles  per  hour. 

From  diagram,   Fig.    19. 

Superelevation  =  5  inches,  k  =  1,  and  L  =  366.7  ft. 

Sta.  P.I  =  21  +21.1 

From  Table  6,  T  =  750.9 

From  Table  1,   o  =  3.58,  o  tan  \l  =     1.7 

From  Table  1,  t  =  183.3 

Ts  =  9       35.9 


Sta.  T.S.  =  11  +85.2 
S  =    3       66.7 


Sta.  S.C.  =  15  +  51.9 
Length  of  circular  curve  =  —   —  -  =     r  —  S  =  10       33.3 


Sta.  C.S.  =  25  +  85.2 
S=    3.      66.7 


Sta.  S.T.  =  29  +  51.9 
Spirals  are  staked  out  either  by  deflection  angles  or  by  offsets. 

82.  Deflection  Angles.  The  process  of  staking  out  spirals  by 
deflection  angles  is  the  same  as  that  for  simple  curves.  Spirals 
rarely  exceed  600  ft.  in  length,  and  the  total  deflection  angle 
to  the  S.C.  rarely  exceeds  5°  00 ',  therefore  the  entire  spiral  can 
be  run  in  from  the  T.S* 

Since  a  spiral  is  more  difficult  to  aline  than  a  simple  curve, 
stakes  should  be  placed  closer  together;  and  since  the  spiral 
becomes  sharper  as  it  increases  in  length,  stakes  should  be 
placed  more  closely  together  at  the  end  than  at  the  beginning. 

*  For  running  in  spiral  from  an  intermediate  point  on  spiral,  see 
paragraph  85. 


CURVES 


57 


RULE: — Place  stakes  50  ft.  apart  on  all  spirals  up  to  the  point 
where  the  degree  of  curve  (d)  becomes  about  3°;  beyond  this  point 
place  them  25  ft.  apart. 

For  ease  in  computation,  stakes  are  placed  the  above  distances 
apart  beginning  at  the  T.S.',  hence  in  general  all  the  stakes  will 
fall  at  plusses. 

The  spiral  deflection  angles  are 


a  (in  minutes)  =  10  ks2 


(20) 


Note  that  the  spiral  deflection  angles  vary  with  the  square  of 
the  distance  instead  of  the  first  power  as  in  simple  curves. 

In  Fig.  21  point  1  is  located  by  a  50-ft.  chord  from  the  T.S. 
and  the  angle  a\\  point  2  by  a  50-ft.  chord  from  1  and  the  angle 


T.S. 


s.c. 


FIG.  21. 


02',  and  so  on,  noting  the  change  in  chord  length  when  d  becomes 
3°.  From  Eq.  20,  and 

ai  =  10'  k  (0.5  )2 

a*  =  10'  k  (1.0  )2  =  4ai 

a3  =  10' k  (1.5  )2  =  9ai 

a4  =  10'k(1.75)2  =  49/4ai 

a5  =  10'  k  (2.0  )2  =  16ai 
A  =  a6  =  10'  k  (2.25)2  =  81/4ai  =  1/3A 

It  is  thus  seen  that  the  deflection  angle  for  the  first  station  only 
needs  to  be  determined  from  Eq.  20.  The  other  values  are 
found  by  multiplying  this  value  by  the  square  of  the  ratio  of  the 
first  distance  to  the  other  distances. 


TRANSIT  NOTES  FOR  LINE  L 

STATION 

ALINE- 
MENT 

TOTAL 
DEFL. 

ANGLE 

CALC. 
BEAR 

MAG. 
BEAR 

REMARKS 

1027 

Tan. 

S  64°-14'W 

S  64°-10'W 

26+23.1 

S.T.      © 

o°-oo' 

+73.1 
25+23.1 
+73.1 

k  =1°- 
in  from  S.T. 

0°-02.o 
0-10 
0-22.5 

24+23.1 
+73.1 

CQ       C3 

0-40 
1-02.5 

+  23.1 

C.S.      0 

1-30 

+23.1 

c.s.    © 

8-45' 

-Ho 

(check) 

23 

8-24 

22 

6-54 

P.I.  1020+40 

21 

.0 

5-24 

1  =  26-30' 

1020 

°co 

3-54 

D=  3-00' 

19 

2-24 

T8=  600  '2 

18 

0-54 

17+39.8 

s.c.     © 

0-00' 

17+39.8 

1-30 

k  -  1- 

+  89.8 

1-02.5 

L=  300.'0 

16+39.8 

7 

0-40 

A  =4-  30' 

+89.8 
15+39.8 

2 

"3. 

CO 

0-22.5 
0-10 

A=  1°-30' 
B  =  3°_00' 

+89.8 

0-02.5 

+39,8 

T.S.      © 

0-00' 

1014 

Tan. 

S  37-  44'\V 

S37°-50'W 

FIG.  22. 


58 


28 
Mike  McCarthy -last. 

Heine  Heinrichson-H.C. 

01ey01sen-R.C.  June  17, 1913 

S.Garibaldi-  R.F.  Clear-Hot 


Set  up  at  S.T.plates  at  0  °00';  B.S.along  Tangent; 
Back  in  Spiral  to  C.S. 


Ic=I-2A=17°-30' 


B.S.  at  T.S.,  Plates  at  3-00'.  Plunge  and  turn  to 
0-00  'for  Tangent. 


FIG.  22.  59 


60  TEXT   BOOK   ON   RAILROAD    SURVEYING 

Example.  Assume  that  in  Fig.  21,  T.S.  is  at  Sta.  711  +  44.0, 
k  =  2,  and  S  =  2.25.  Then  D  =  4°  30'. 

Deflection  angle  of  T.S.,  Sta.  711  +  44.0  =  0°  00' 

"      "     1        "    711  +94.0  =  0°05' 

"  "       "     2        "    712+44.0  =  0°20' 

"       "     3        "    712  +  94.0  =  0°  45' 

"       "     4        "    713  +  19.0  =  1°01' 

(1°  01. 25  'exact) 

"  "       "     5       "    713  +  44.0  =  1°  20' 

"       "S.C.(A)"    713  +  69.0  =  1°41' 

(1°  41. 25' exact) 
A  -  1/3A,  A  =  5°-04' 

(5°  03.75'  exact) 
B  =  2/3A  =  2A     =3°  22.5' 

To  orient  the  transit  at  the  S.C.,  back-sight  on  the  T.S.  with 
the  plates  set  at  3°  22.5'  (angle  B)',  then  turn  the  plates  to  zero 
and  the  telescope  is  on  tangent  and  the  circular  curve  is  run  in 
as  from  the  T.C.  of  a  simple  curve,  stakes  being  placed  at  the 
regular  stations  (in  the  above  problem  at  714  +  00,  714  +  50,  etc.). 
After  running  in  about  half  of  the  circular  curve,  move  the 
transit  to  the  S.T.  and  run  in  the  second  spiral  with  the  same 
deflection  angles.  Then  move  to  the  C.S.  and  back-in  the 
remainder  of  the  circular  curve,  thus  placing  the  adjustment 
of  the  errors  of  surveying  at  the  center  of  the  curve  instead  of  at 
the  end  of  the  spiral  as  is  usually  but  unwisely  done.  The  form 
of  notes  is  shown  in  Fig.  22. 

This  method  is  particularly  applicable  for  locating  spirals 
after  the  construction  is  completed  and  the  track  is  to  be  brought 
to  exact  line  and  surface.  For  setting  stakes  during  construction 
the  offset  method  is  preferable. 

83.  Offsets.  Evidently  the  entire  spiral  can  be  located  by 
means  of  the  coordinates  x  and  y  (tangent  offsets),  and  this  is  a 
satisfactory  method  for  short  flat  spirals  where  y  is  less  than 
about  10  feet.  When  y  becomes  greater  than  this,  the  spiral 
can  not  be  located  with  sufficient  accuracy  unless  the  offsets  are 
turned  off  with  an  instrument. 

On  location,  since  it  is  usually  desirable  to  advance  the  line 
as  rapidly  as  possible,  the  best  method  is  to  run-in  the  circular 
curve  from  the  T.C.  to  the  C.T.  (offsetted  curve,  see  Fig.  20), 


CURVES  61 

and  to  insert  the  spiral  later  by  offsets  from  both  the  tangent 
and  the  circular  curve. 

Since  the  spiral  departs  from  an  osculating  circle  at  any  point 
at  the  same  rate  that  it  departs  from  the  tangent  at  the  T.S. 
(see  Appendix,  paragraph  142),  it  follows  that  the  offsets  from 
the  circular  curve  to  the  spiral  are  the  same  as  the  offsets  from 
the  tangent  at  the  T.S.  for  the  same  distances.  Since  the 
offset  o  and  the  spiral  bisect  each  other  (see  Appendix,  para- 
graph 141),  it  is  evident  that  the  maximum  offset  is  Jo,  and  there- 
fore will  be  small,  and  also  that  it  is  necessary  to  compute  offsets 
for  half  the  spiral  only.  Half  of  the  spiral  is  then  located  by 
offsets  y  from  the  tangent  at  the  T.S.,  and  the  other  half  by  the 
same  offsets  measured  normal  to  the  circular  curve. 

From  Eq.  22  it  is  seen  that  the  offsets  vary  approximately  as 
the  cube  of  the  distances,  and  for  this  method  of  location  can  be 
taken  so  with  inappreciable  error.  Since  the  maximum  ordinate 
is  £o,  and  o  is  determined  to  offset  the  T.C.,  it  is  seen  that  the 
offsets  can  be  determined  directly  from  o  instead  of  from  Eq.  22. 

Application  to  Compound  Curves 

At  the  C.C.  of  a  compound  curve — as  at  the  T.C.  of  a  simple 
curve — there  is  a  change  in  the  rate  of  curvature  and  in  the 
amount  of  superelevation,  and  if  this  is  great  enough  to  be  ob- 
jectionable, a  spiral  should  be  inserted  between  the  two  branches. 

Evidently  only  that  part  of  the  spiral  of  curvature  intermediate 
between  the  degrees  of  the  two  curves  is  required,  and — as  at  the 
T.C.  of  a  simple  curve — the  two  curves  must  be  offsetted  at  the 
C.C. 

84.  Problem  1.  In  Fig.  23,  HBC  is  a  A  curve  and  EFJ 
is  a  D2  curve  having  parallel  tangents  at  C  and  E  (the  position 
of  the  C.C.  if  the  curve  were  unspiraled).  It  is  desired  to  con- 
nect the  two  curves  by  the  spiral  BF.  Consider  the  spiral  run 
backwards  to  its  T.S.  at  A.  Since  the  degree  of  curve  of  the 
spiral  at  B  must  be  A  and  at  F  must  be  Z)2,  then  the  spiral  from 
B  to  F  is  that  part  of  a  regular  spiral  from  where  d  =  Di  to 
where  d  =  D2.  The  value  of  k  depends  upon  the  maximum 
speed  permissible  on  the  sharper  curve  and  upon  the  difference 
in  the  two  degrees  of  curve. 

Since  the  spiral  departs  from  every  osculating  circle  at  the 


62 


TEXT   BOOK    ON   RAILROAD    SURVEYING 


same  rate  as  from  the  tangent  at  the  T.S.,  BC  =  CF,  the  spiral 
bisects  CE,  and  CE  is  equal  to  o-for  a  spiral  whose  D  =  D2  —  D\ 
for  the  chosen  value  of  k. 


FIG.  23. 

To  insert  a  spiral  between  the  curves,  find  o  for  a  Z>2  —  A 
curve  and  make  the  offset  from  C  to  #.  The  length  of  the  spiral 
in  stations  is 

D2  -  D! 


Locate  B  and  F  by  measuring  J$i  from  C  and  E. 

The  curves  may  be  staked  out  by  continuing  the  first  branch 
to  C,  offsetting  to  E,  and  running-in  the  second  branch;  and 
then  inserting  the  spiral  by  offsets  from  the  circular  curves  in  ex- 
actly the  same  way  as  explained  in  paragraph  83. 

85.  Deflection  angles  may  also  be  used;  but,  since  the  transit 
is  at  a  point  on  the  spiral,  B  or  F,  and  not  at  the  T.S.,  the  reflec- 
tion angles  to  be  used  are  values  of  /  instead  of  a.  From  a 
tangent  at  B  in  Fig.  23,  the  deflection  angle  n  to  any  point 
N'  on  the  circular  curve  BCM  is  JA  X  BN'.  Since  the  spiral 
departs  from  the  osculating  circle  at  the  same  rate  as  from  the 
tangent  at  the  T.S.,  the  angle  between  the  circular  curve  and  a 
point  N  on  the  spiral  is  a  =  1/6  k(BN)*.  But  BN  and  BN' 
are  equal.  Then 


f  =  n  +  a  = 


(BN)  +  1/6  k  (BN)2. 


CURVES  63 

If  the  transit  were  set  at  F,  the  deflection  angle  to  N  would  be 
f  =  JD2  (FN)  -  1/6  k  (FN)2 

86.  Problem  2.     Fig.    24   shows   a   compound   curve   to   be 
spiraled   at   each   end   and   between   the   two   branches.     The 


FIG.  24. 

tangent  distances  Tsi  and  TS2  are  required,  and  the  solution  is 
fully  indicated  in  the  figure. 

Application  to  Existing  Curves 

To  insert  a  spiral  in  an  existing  track,  it  is  necessary  to  shift 
the  ends  of  the  curve  inward  to  provide  room  for- the  spiral. 
If  the  same  degree  of  circular  curve  were  retained,  this  would 
shift  the  entire  curve  inward.  Since  the  amount  of  such  shifting 
may  be  considerable  (o  sec.  JJ  at  its  center)  the  new  alinement 
may  not  be  on  the  old  roadbed  and  considerable  cost  of  earth- 
work would  be  entailed  in  making  the  change.  To  obviate  this, 
the  degree  of  curve  may  be  changed  in  such  a  way  that  the  new 
alinement  will  permit  the  insertion  of  the  spiral  and  at  the  same 
time  require  little  or  no  additional  earthwork. 


64 


TEXT   BOOK    ON   RAILROAD    SURVEYING 


87.  SIMPLE    CURVES.     Existing    simple    curves    may    be 
spiraled  in  two  ways. 

(1)  By  shifting  the   center  of  the  curve  outward  *  a   small 
amount,  and  by  sharpening  the  curve  sufficiently  to  give  the 
desired  offset.     This  method  is  particularly  applicable  for  curves 
whose  lengths  are    less   than   about  four  times  the  length  of 
the  spiral  to  be  used. 

(2)  By  sharpening  the  curve  slightly  at  the  ends  to  give  the 
necessary  offset,  leaving  the  middle  portion  of  the  curve  un- 
changed.    This  method  is  preferable  for  long  curves. 

88.  First  Method. 

In  Fig.  25,  ABC  is  the  existing  unspiraled  curve  with  B  as 
its  middle  point.     It  is  desired  to  shift  the  track  outward  at  B 


FIG.  25. 

the  assumed  distance  p,  and  to  sharpen  the  curve  to  give  the 
necessary  offset  o. 

From  the  figure,  p  is  the  difference  between  the  external  dis- 
tances of  the  existing  curve  and  the  spiraled  curve.  Therefore 
from  Eqs.  5  and  27,  we  have 

p  =  R  exsec  JI  —  [(Ri  -f  o)exsec  JI  +  o] 

=  (R  -  Ri  -  o)  exsec  JI  -  o      ...     (32) 

*The  shifting  must  be  outward  to  make  the  change  in  alinement  a 
minimum. 


CURVES  65 

The  minimum  change  in  radius  (R  —  Ri)  is  that  which  will  make 
p  equal  to  zero.  Then  from  Eq.  32, 

Ri  =  R ^T-  .  (33) 

vers  il 

(This  is  the  maximum  value  of  Ri  that  will  satisfy  the  con- 
ditions.) 

In  any  given  problem  first  determine  approximately  (mentally) 
the  maximum  value  of  Ri  from  Eq.  33.  This  is  used  simply  as  a 
guide  in  estimating  the  value  of  RI  to  use.  Choose  a  value  of 
Ri  less  than  this  (preferably  to  agree  with  an  even  10  minutes  of 
degree  of  curve)  and  solve  Eq.  32  for  p.  If  this  value  agrees 
sufficiently  close  with  the  desired  shifting  of  the  track  at  Bt  the 
solution  is  complete.  If  not,  re-estimate  Ri  (using  the  previous 
value  as  a  guide),  and  repeat  until  a  satisfactory  value  of  p 
is  obtained. 

The  distance  from  the  T.C.  of  the  original  curve  to  the  T.S. 
is  the  difference  between  Ts  for  the  new  curve  and  T  for  the  old 
curve.  The  T.S.  and  the  S.T.  are  then  located  from  the  T.C. 
and  the  C.T.,  respectively,  of  the  old  curve,  and  the  new  curve  is 
staked  out  as  in  new  work. 

Example.  /  =  40°  00',  D  =  4°  00',  k  =  1,  and  p  =  about 
1.5  ft. 

From  Eq.  33,  the  maximum  value  of  Ri  =  1330  ft.  (approx.) 
corresponding  to  DI  =  4°  18'  (approx.).  Choose  a  trial  value  of 
Ri  =  1322.2  ft.  for  A  =  4°  20'.  Then  from  Eq.  32,  p  =  0.8  ft. 
Choosing  Ri  =  1312.1  ft.  for  A  =4°  22',  p  becomes  1.3  ft.,  showing 
that  a  change  of  2'  in  D!  increases  p  about  0.5  ft.  Then  to  make 
p  =  1.5  ft.,  DI  would  have  to  be  about  4°  23'.  Since  an  odd 
value  of  DI  is  undesirable,  and  since  it  is  not  necessary  to  make 
p  exactly  1.5  ft.,  it  is  sufficient  to  take  DI  =  4°  22'. 

In  the  foregoing  discussion  it  was  assumed  that  the  degee  of 
the  existing  curve  was  known  and  that  the  T.C.  and  C.T.  were 
monumented.  It  will  generally  happen,  however,  that  the 
degree  of  curve  is  unknown,  that  the  track  is  in  poor  alinement, 
and  that  the  T.C.  and  C.T.  are  not  monumented.  In  this  case, 
run  out  the  tangents  to  an  intersection  and  measure  the  inter- 
section angle.  Then  measure  the  external  distance  of  the 


*o  should  correspond  with  Ri  which  is  unknown,  therefore  use 'a 
value  of  o  corresponding  to  about  0.9  R. 


66  TEXT   BOOK   ON   RAILROAD   SURVEYING 

existing  curve  and  from  it  determine  the  degree  of  the  curve  that 
will  connect  the  tangents  and  pass  through  the  middle  point  of 
the  existing  curve.  Use  the  value  of  R  corresponding  to  this 
value  of  D  in  solving  Eqs.  32  and  33. 

89.  Second  Method.  Case  1.  In  Fig.  26,  ABC  is  the  exist- 
ing unspiraled  curve,  whose  degree  of  curve,  D,  is  known,  whose 
alinement  is  good,  and  whose  T.C.  and  C.T.  are  monumented. 
At  some  point  on  this  curve,  such  as  B,  it  is  desired  to  compound 
with  an  assumed  curve  of  slightly  shorter  radius,  Ri,  which, 
when  run  to  a  tangent  parallel  to  the  initial  tangent,  will  be  a 
distance  from  it  equal  to  o  for  a  Ri  curve,  thus  providing  room 
for  the  spiral. 

In  the  figure,  EFB  is  the  Ri  curve,  and  EG  is  o  corresponding  to 


T.S.       A  T.C.  G 


FIG.  26. 

it.     It  is  required  to  find  the  location  of  the  T.S.,  of  the  S.C., 
and  of  the  C.C.,  B. 
From  the  figure, 

o  =  EG  =  GL  -  EL  =  (R  -  Rx)  vers  Ii 

vers  Ii  =  ^  _  R (34) 

The  T.S.  is  located  by  measuring  back  from  the  old  T.C. 

T.C.  to  T.S.  =  t  -  (R  -  Ri  -  o)  tan  Ii.     .     .     (35) 

The  S.C.  is  located  by  measuring  the  spiral  length  from  the  T.S. 
The  C.C.,  J5,  is  located  by  measuring  the  distance  FB  from  the 

S.C. 

FB  (in  feet)  =  --^   10°       •     -     •     •     (36) 


CURVES 


67 


The  location  of  the  point  B  may  be  checked  by  measuring  the 
distance  AB  along  the  old  curve. 

AB  (in  feet)  =  -g-  100 

The  distance  from  the  T.S.  to  the  C.C.  along  the  new  aline- 
ment  is  shorter  than  along  the  old  alinement  which  will  require  the 
rails  to  be  cut.  This  shortening  is  equal  to  the  difference  in  the 
Sta.  numbers  of  the  C.C.  as  computed  along  the  respective 
alinements. 

The  limits  of  RI  are  such  as  will  make  the  point  B  come  at 
the  middle  of  the  original  curve,  or  will  make  B  and  F  coincide. 
In  general,  R\  should  be  chosen  between  0.8  R  and  0.9  R. 

Example.  D  =  3°  00',  A  (assumed)  =  4°  00',  k  =  1,  and 
o  =  4.65.  From  Eq.  34,  h  =  12°  14.3'.  From  Eq.  35,  the 
distance  from  the  old  T.C.  to  the  T.S.  is  156.5  ft.  The  spiral 
is  400  ft.  long.  From  Eq.  36,  the  distance  from  the  S.C.  to  the 
C.C.  is  121.1  ft. 

90.  Second  Method.     Case  2.     In    Fig.    27,    ABC    is    the 

existing  unspiraled  curve,  whose  degree  of  curve  is  unknown, 
whose  alinement  may  be  poor,  and  whose  T.C.  and  C.  T.  are  no*, 
monumented.  First  set-up  in  the  center  of  the  track  at  some 


_A_     G ,  O          (P         L 
--^F^fec^— -TN~~T 
V/LB 


FIG.  27. 

point  C  near  the  middle  of  the  curve  and  by  trial  deflection 
angles  find  the  degree  of  the  curve,  D,  that  will  most  nearly 
conform  to  the  existing  track.  Then  run-in  this  D  curve  to  a 
point  B  which  is  about  500  ft.  from  the  end  of  the  curve.  If  this 
curve  were  continued,  its  T.C.  would  fall  at  /  on  a  tangent 
parallel  to  the  initial  tangent  QL  and  at  a  distance  p  from  it. 


68 


TEXT   BOOK   ON   RAILROAD    SURVEYING 


At  B  run-out  the  tangent  to  the  D  curve  to  an  intersection 
with  the  initial  tangent  at  L  and  measure  the  intersection  angle 
IB  and  the  distance  BL.  BN  is  the  tangent  distance  of  the 
curve  corresponding  to  IB-  Then, 

p  =  NL  sin  IB 

The  D  curve  is  to  be  compounded  with  a  chosen  D\  curve  at 
some  point  F  to  give  the  required  offset  o. 


EK  =  o  —  p  =  (R  —  Ri)  vers  IB,  whence 
o  —  p 


(37) 


The  T.S.  is  located  at  Q  by  measurement  from  L. 

QL  =  QG  +  GL  =  t  +  JN  -  JK  -  NL  cos  IB 

The  S.C.  and  the  C.C.  are  located  as  in  Case  1,  and  the  spirals 
are  staked  out  as  in  new  work. 

J  may  fall  outside  of  QL,  in  which  case  p  and  o  are  numerically 
added;  again,  p  may  be  greater  than  o,  and  in  this  case  Ri  must 
be  greater  than  R. 

91.  COMPOUND  CURVES. 

An  existing  compound  curve  can  be  spiraled  in  the  following 
manner. 


FIG.  28. 


In  Fig.  28,  ABC  is  the  existing  compound  curve  with  its  C.C. 
Sit  B.    To  make  room  for  the  spiral  it  is  necessary  to  compound 


CURVES  69 

the  sharper  branch,  D2,  at  some  point  C  with  a  still  sharper 
curve  of  chosen  degree,  D3,  and  to  continue  the  first  branch, 
DI,  to  the  point  E  where  its  tangent  is  parallel  to  the  tangent  of 
the  D3  curve  at  G.  EG  is  the  offset  o  for  a  spiral  whose  D  =  D3 
-  DI.  The  problem  is  to  find  BE  =  HI  and  CG  =  n2. 

For  small  central  angles  the  ordinates  between  two  circular 
curves  are  equal  to  the  difference  in  their  tangent  offsets.  There- 
fore EF  can  be  considered  as  the  difference  in  the  tangent  offsets 
of  the  DI  and  D2  curves  in  the  distance  HI]  similarly  FG  is  the 
difference  in  the  tangent  offsets  of  the  D2  and  D3  curves  for  the 
distance  n2. 

Then  from  Eq.  15, 

o  =  EF  +  FG  =  0.873  (D2  -  DI)  ni2  +  0.873  (D3  -  D2)  n22. 

Since  the  central  angle  of  the  arc  EC  equals  the  sum  of  the  cen- 
tral angles  of  the  arcs  BE  and  CG, 

D2  (ni  +  n2)  =  Dini  +  D3n2 
or  (D2-  DI)  ni  =  (D3-  D2)  n2. 

Solving  these  simultaneous  equations  for  HI  and  n2  we  have 


n     -   1  07      I  (D3  -  P2)  O  /OON 

L07\'  (D2  -  Dl)  (Da-DJ- 

n,  =  1  07     !~~~      (E>2  -  Di)o~  /oqi 

\(D3-D2)(D3-D1) 


C  is  located  by  measuring  the  distance  n\  +  nz  along  the  D2 
curve  from  B.  The  C.S.  is  located  by  running  the  DI  curve 
from  B  the  distance  HI  —  t,  and  the  S.C.  is  located  by  running 
the  D3  curve  from  C  the  distance  HZ  —  (Si  —  t).  The  spiral  is 
staked  out  as  usual  for  compound  curves. 


70 


TEXT   BOOK    ON   RAILROAD    SURVEYING 


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CURVES 


71 


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72  TEXT   BOOK   ON   RAILROAD   SURVEYING 

Problems 

1.  What  is  the  maximum  speed  for  10°,  7°,  3°,  and  2°  curves? 

2.  Given.     P.I.  at  Sta.  741  +  60.0;  /  =  35°  42';  k  =  2°;  and 
D    =    5°  00'.     Determine  the  station  numbers  of  the    T.S., 
S.C.,  C.S.,  and  S.  T.,  and  the  external  distance  of  the  curve. 

Answer.  T.S.  =  736  +  65.3,  S.C.  =  739  +  15.3,  C.S.  = 
743  +  79.3,  S.T.  =  746  +  29.3. 

3.  Write  transit  notes  for  Problem  4. 

4.  Compute  notes  for  locating  the  spiral  in  Problem  4  by 
offsets. 

-5.  On  new  location  a  5°  00 '  curve  is  to  be  offsetted  from  a 
2°  00'  curve  at  station  333  +  00.0  f or  a  k  =  1°  spiral.  Find  the 
station  numbers  of  the  C.S.  and  the  S.C.  and  the  required  offsets. 
Write  notes  for  spiral. 

6.  Write  notes  to  locate  spiral  in  Problem  7  by  offsets. 

7.  Given.     P.I.  at  Sta.  267  +  00.0;  /  =  69°  20';  /i   =  21° 
40';  72  =  47°  40';  A  =  1°  30';  A  =  4°  00';  ki  =  /c2  =  1°; 
/c3=2°.     Find  TA,  TS2,  and  the  Sta.  numbers  of  the  T.S.,  S.C., 
C.S.,  S.C.,  C.S.,  and  S.T. 

Answer.  Tsl  =  1883.3,  T&  =  1272.3,  T.S.  =  248  +  16.7, 
S.C.  =  249  +  66.7,  C.S.  =  262  +  11.1,  S.C.  =  264  +  61.1, 
C.S.  =  274  +  27.8,  S.T.  =  276  +  27.8. 

8.  Write  transit  notes  for  Problem  9. 

11.  An  existing  curve  is  to  be  spiraled.  7  =  39°  18',  D  =  3° 
20',  k  =  1°.  The  track  is  to  be  shifted  outward  at  the  middle 
about  1.0  ft.  Find  A  and  the  actual  value  of  p. 

Answer.     A    =  3°  30';  p   =   1.24  ft. 

9.  Given  an  existing  2°  40'  curve  to  be  spiraled  by  compound- 
ing near  the  ends  with  a  3°  00'  curve.     The  T.C.  is  at  Sta. 
66  +  66.6  and  k  =  1°.     Find  the  station  number  of  the  T.S., 
S.C.,  and  C.C.,  and  the  amount  that  the  track  will  be  shortened. 

Answer.  T.S.  =  65  +  47.1,  S.C.  =  68  +  47.1,  C.C.  =  69 
+  42.0.  The  track  is  shortened  0.1  ft. 

10.  An  existing  compound  curve  has  its  C.C.  at  Sta.  488+ 
50.0.  A  =  4°  00',  A  =  7°  00',  A  =  7°  40',  and  k  =  1°.  Find 
the  station  numbers  of  the  C.S.,  S.C.,  and  C.C. 

Answer.  C.S.  =  487  +  16.5,  S.C.  =  490  +  83.3,  C.C.  = 
491  +  24.3. 


CHAPTER  V 

EARTHWORK 

Introduction 

92.  Railroad  Cross -Sections.  The  forms  of  earthwork  cross- 
sections  in  cut,  fill,  and  side-hill  work  (both  cut  and  fill)  are 
shown  in  Figs.  29,  30,  and  31,  respectively.  Fig.  32  shows  the 
form  a  section  in  cut  should  take  when  it  consists  partly  of  earth 
and  partly  of  rock. 


FIG.  29. 


FIG.  30. 


FIG.  31. 


FIG.  32. 


93.  Width  of  Roadbed.  The  width  of  roadbed  for  single  track 
on  fill  varies  from  14  to  22  feet,  and  in  cut  from  20  to  33  feet. 
However,  the  best  practice  is  18  or  20  feet  on  fill  and  28  or  30 
feet  in  cut.  The  greater  width  in  cut  is  required  to  allow  for 
side  ditches  for  drainage. 

For  double  track  add  to  the  above  quantities  the  distance 
between  track  -centers. 

Copyright,  1913,  by  G.  W.  Pickels  and  C.  C.  Wiley. 

73 


74  TEXT    BOOK   ON    RAILROAD    SURVEYING 

94.  Ditches.     In    order    to    provide    drainage    in    cuts    side 
ditches  are  constructed  on  each  side  of  the  track.     The  ditches 
should  be  of  such  cross-section  that  they  can  be  easily  constructed 
and  maintained  and  not  easily  obstructed.     For  these  reasons 
broad  shallow  ditches  are  better  than  narrow  deep  ones.     The 
usual  width  of  ditches  is  from  3  to  6  feet,  depending  on  the  size 
of  the  cut  and  the  amount  and  rate  of  rainfall. 

Where  there  is  considerable  ground  water  it  may  be  necessary 
to  construct  tile  sub-drains  under  one  or  both  of  the  side  ditches. 

When  there  is  likelihood  of  considerable  surface  water  draining 
on  to  the  roadbed,  as  for  example  in  side-hill  work,  an  open 
diversion  ditch,  termed  a  surface  ditch,  should  be  constructed 
outside  the  cut  to  carry  this  water  to  the  end  of  the  cut. 

Side  ditches  are  more  expensive  to  excavate  than  the  body  of 
the  cut,  and  therefore  should  preferably  be  included  as  a  separate 
item  in  the  contract. 

95.  Side  Slopes.     The  slope  of  the  sides  of  the  cut  or  fill  is 
expressed  as  the  ratio  of  the -horizontal  to  the  vertical  distance 
and  depends  on  the  material  through  or  of  which  they  are  made. 
The  following  are  commonly  used. 

(1)  Cut 

Solid  rock  -  vertical  or  J:l. 

Loose  rock  —  J:l    to  1:1. 

Hard  earth  -  1:1    to  11:1. 
Soft  earth 

or  Sand  -  1J:1  to  3:1. 

(2)  Fill 

Loose  rock    —  1:1. 

Earth  -1:1    to  11:1. 

Sand  -  11:1  to  2:1. 

96.  Borrow  Pits.     It  frequently  happens  that  the  earth  from 
the  adjacent  cuts  is  not  sufficient  to  make  the  intervening  fill. 
In  this  case  the  contractor  is  allowed  to  borrow  earth  from  along 
the  right  of  way.     The  pits  thus  dug  are  called  borrow  pits,  and  are 
staked  out  by  the  engineer  before  the  work  is    started.     In 
staking  out  borrow  pits  care  should  be  taken  to  leave  a  berm 
of  at  least  5  ft.  between  the  foot  of  the  embankment  and  the 


EARTHWORK  75 

edge  of  the  borrow  pit;  a  similar  berm  should  also  be  left  at  the 
edge  of  the  right  of  way.  The  contractor  is  required  to  leave 
all  borrow  pits  in  such  a  shape  that  water  will  not  collect  in 
them. 

97.  Shrinkage.     Some    soils    become    more    compact    when 
thoroughly    compressed    in    an    embankment   than    they  were 
originally,  and  therefore  a  cubic  yard  of  cut  will  ultimately  make 
less  than  a  cubic  yard  of  fill.     This  does  not  take  place  until 
a  year  or  two  after  the  fill  is  made.     This  reduction  of  volume 
is  called  shrinkage.     The  amount  of  shrinkage  varies  with  the 
kind  of  soil  and  hence  the  allowance  for  shrinkage  should  be  left 
to  the  judgment  of  the  engineer  in  the  field  and  should  not  be 
made  part  of  an  arbitrary  set  of  standards.     Very  few  soils  will 
shrink  as  much  as  3  per  cent. 

98.  Growth.     Solid  rock  will  increase  in  volume  on  excavation 
and  hence  a  cubic  yard  in  cut  will  make  more  than  a  cubic  yard 
of  fill.     This  is  termed  growth.     The   amount   of  growth   de- 
pends on  the  uniformity  of  size  of  the  fragments,  and  varies  from 
30  to  100  per  cent.,  the  average  in  railroad  work  being  about 
50  per  cent. 

99.  Settlement.     A  newly  built  fill  is  never  thoroughly  com- 
pact and  therefore  will  continue  to  settle  for  some  time,  possibly 
a  year  or  two.     The  difference  in  height  between  the  newly- 
built  fill  and  the  fill  after  fully  settling   is  termed  settlement. 
The  amount  of  settlement  depends   on  the  nature  of  the  soil 
and  the  method  of  forming  the  embankment.     A  fill  of  ordinary 
soil,  well  made  with  horses  and  scrapers  or  wagons,  will  settle 
about  5  per  cent.;  if  simply  dumped  from  a  trestle,  the  settle- 
ment is  greater  and  in  some  cases  may  reach  15  per  cent.     A 
rock   fill   will   settle   but    little. 

Allowance  for  settlement  is  made  in  the  field  by  marking  on  the 
slope  stake  the  theoretical  fill  plus  the  allowance  for  settlement. 
For  example,  if  the  fill  is  10  feet  and  the  settlement  is  5  per  cent.; 
the  slope  stake  is  set  at  the  proper  point  for  a  fill  of  10  feet, 
but  is  marked  F  10.5. 

Shrinkage  and  settlement  are  frequently  confused,  but  it  is 
to  be  noted  that  settlement  includes  shrinkage,  and  that  the 
allowance  in  the  field  is  for  settlement. 


76  TEXT   BOOK    ON    RAILROAD    SURVEYING 

If  a  fill  is  short  and  high  the  full  allowance  for  settlement  may 
make  a  hump  in  the  grade  line  which  may  be  very  objectionable 
from  the  standpoint  of  operation.  Of  course  this  hump  would 
ultimately  disappear,  but  it  may  be  preferable  to  avoid  the  hump 
by  reducing  the  allowance  for  settlement  and  later  to  bring  the 
track  up  to  grade  by  the  addition  of  ballast. 

In  case  a  bridge  is  placed  in  a  high  fill,  the  full  allowance  for 
settlement  can  not  be  made  at  this  point.  Extra  care  should  be 
taken  to  compact  the  fill  immediately  behind  the  abutments  in 
order  to  reduce  the  amount  of  settlement,  and  the  track  should 
be  carefully  maintained  at  these  points  until  the  embankment 

is  fully  settled. 

» 

Cross-Sectioning 

100.  Slope  Stakes.  Before  construction  work  can  be  com- 
menced, slope  stakes  must  be  set  for  the  guidance  of  the  con- 
tractor (1)  at  every  full  station,  (2)  at  those  intermediate  points 
where  the  longitudinal  slope  of  the  ground  changes,  and  (3) 
at  other  points  required  by  special  structures. 

A  slope  stake  is  a  stake  that  is  set  on  each  side  of  the  center 
line  at  the  point  where  the  side  slope  of  the  cut  or  fill  will  inter- 
sect the  ground  surface.  It  has  the  cut  or  fill  at  that  point  (see 
settlement)  marked  on  one  side  and  the  station  number  on  the 


FIG.  33. 

* 

other.  The  numbers  are  written  lengthwise  of  the  stake  be- 
ginning at  the  top,  and  the  cut  or  fill  is  given  in  feet  and  tenths, 
prefixing  C  for  cut  and  F  for  fill  (see  Fig.  33).  The  stakes  are 
driven  with  the  tops  slanting  outward,  and  the  sides  upon  which 
the  cuts  or  fills  are  marked  facing  the  roadbed,  as  shown  in 
Fig.  33. 


EARTHWORK  77 

101.  Cross-Sectioning.      Cross-sectioning  consists  in  setting 
the  slope  stakes,  which  mark  the  limits  of  excavation  or  embank- 
ment, and  in  taking  sufficient  elevations  between  the  slope  stakes 
to  compute  the  area  of  the  section  to  be  excavated  or  filled  in. 

The  data  needed  for  cross-sectioning  are  the  sub-grade  eleva- 
tion at  every  station,  the  width  of  the  roadbed  in  both  cut  and 
fill,  the  side  slopes  for  different  materials,  and  the  elevations  of 
all  bench  marks.  The  sub-grade  elevation  for  every  station  on 
the  entire  line  can  be  obtained  from  the  profile  and  should  be 
written  in  the  cross-section  notebook  opposite  its  station — as 
shown  in  the  form  of  notes  in  Fig.  34 — and  carefully  checked 
before  any  cross-sectioning  is  done.  Also  the  description  and 
elevation  of  every  bench  mark  that  was  established  during  the 
location  should  be  written  in  the  back  of  the  notebook. 

102.  Field  Work.     Starting   with   the   nearest   bench   mark, 
differential  levels  are  run  to  the  part  of  the  line  that  is  to  be 
cross-sectioned.      The  back   sights,   fore  eights,   heights  of  in- 
strument,  and  elevations  of   turning  points  are  recorded  in  the 
second  column  of  the  notes  in  such  a  way  that  the  H.I.  will 
be  in  the  same  line  as  the  first  station  cross-sectioned  from  that 
set-up  (see  Fig.  34) .     The  sub-grade  elevation  in  the  third  column 
is  subtracted  algebraically  from  the  H.I.  elevation  in  the  second 
column  and  the  difference  is  the  grade  rod  and  is  recorded  in  the 
fourth  column.     A  rod  reading  is  then  taken  to  the  nearest  tenth 
of  a  foot  on  the  ground  at  the  foot  of  the  center  stake.     All  the  rod 
readings  are  considered  negative.     The  rod  reading  and  the  grade 
rod  are  added  algebraically  and  the  sum  represents  the  cut  or 
fill  at  the  point  where  the  reading  is  taken,  minus  indicating  fill 
and  plus  indicating  cut.     The  cut  or  fill  is  marked  on  the  back 
of  the  center  stake  and  is  recorded  in  the  notes  as  the  numerator 
of  a  fraction  whose  denominator  is  zero. 

If  the  ground  is  level  transversely  of  the  line,  the  cut  or  fill 
at  the  slope  stake  will  be  the  same  as  at  the  center,  and  the 
distance  from  the  center  stake  to  the  slope  stake  is  found  by 
multiplying  the  center  cut  or  fill  by  the  ratio  of  the  side  slopes 
and  adding  one  half  the  width  of  the  roadbed.  Such  a  section 
is  called  a  level  section  (see  Fig.  35a). 

If  the  ground  is  not  level  transversely  the  cut  or  fill  will  be 
different  for  various  points,  and  the  problem  is  to  find  the  point 
on  each  side  of  the  center  line  whose  distance  from  the  center  is 


CROSS-SECTION  NOTES. 


Station. 

Turning 
Points. 

Grade. 

Grade 
Rod. 

L. 

C. 

R. 

55 

56 
57 
58 
59 

10560 
61 
62 
63 
64 
65 
66 

740.51 
-11.63 

727.50 

727.00 
726.50 
726.00 
725.50 

725.00 
724.50 
724.00 
723.50 

723.00 
^  > 
722.50 

m  m  > 
722.00 

13.0 

13.5 
14.0 
14.5 
4.1 

4.6 
5.1 
5.6 
6.1 
6.6 

Vs 

^0.3 

+9.0 

+9.8 

+  10.2 

27.5 

+7.3 

0 

+7.3 

29.3 

+7.8 

728.88 
+0.72 

729.60 
-10.54 

25.0 

+5.5 

0 
+6.0 

25.7 

+6.4 

22.3 

+4.2 

0 
+4.4 

23.6 
+4.4 

20.3 

+2.2 

0 

+2.7 

20.6 

+  3.4 
19.1 

+  1.4 

17.3 

+  1.0 

0 

+  1.2 

719.06 
+2.53 

721.69 

15.5 
Grade 
-1.2 

0 
Point  +65 
-0.6 

16.1 

-  0.8 

10.8 
-3.2 

0 
-2.4 

10.2 
-  2.1 

13.8 
-4.8 

0 
-3.8 

12.2 
-  3.5 

16.2 
-6.0 

0 
-5.2 

14.3 

-  5.0 
16.5 

-  6.6 

18.0 
-6.8 

0 
-6.6 

19.2 

-8.5 

0 

-7.8 

18.9 

-  7.0 
19.5 

21.8 

0 

FIG.  34. 


78 


End  Areas. 

Volumes. 

Mass 
Curve 
Volumes. 

Cut. 

Fill. 

Cut. 

Fill. 

412.7 

3653 

1303 

290.8 

2350 

221.0 

948 

1402 

150.2 

687 

715 

88.3 

442 

273 

230 

35.8 
0.0 

0.0 
15.3 

43 

10 

43 
00 

55.0 

130 

10 
140 

278 

95.3 

418 

434 

139.2 

852 

602 

186.0 

1454 

230.8 

772 

2226 

FIG.  34. 


79 


80 


TEXT   BOOK    ON    RAILROAD    SURVEYING 


equal  to  the  cut  or  fill  at  that  point  multiplied  by  the  slope  ratio 
and  added  to  one-half  the  roadbed.  This  is  necessarily  a  cut- 
and-try  method,  but  proficiency  is  soon  attained  in  approxi- 
mating the  correct  position  of  the  slope  stak'^,  and  it  is  seldom 
that  more  than  three  trials  are  required.  The  cut  or  fill  is  marked 


(a)  Level  Section 


( t>)  Three-Level  Section 


(c)  Five-Level  Section 


(d)  Irregulkr  Section 


FIG.  35. 


FIG.  36. 


on  the  slope  stake  and  is  recorded  in  the  notebook  as  the  numera- 
tor of  a  fraction  whose  denominator  is  the  distance  out  from  the 
center.  Three-level,  jive-level,  and  irregular  sections  come  under 
this  heading  (see  Figs.  35b-c-d). 

103.  Sections  at  Grade  Points.  Fig.  36  shows  the  junction 
between  a  cut  and  a  fill.  Stakes,  marked  "  grade,"  are  driven 
at  A,  B,  C,  and  D,  which  are  grade  points.  A  cross-section  is 
taken  at  B  and  D,  and  the  distances  x  and  y  are  measured. 


EARTHWORK 


81 


Areas  of  Cross-Sections 

Before  the  volume  of  earthwork  can  be  computed,  the  area 
of  each  cross-section  must  be  found.     This  is  done  as  follows: 


FIG.  37. 

The  cuts  at  A,  B,  C,  E,  F,  and  G  (i.e.,  the  vertical  distances 
from  the  ground  surface  to  the  level  of  the  roadbed),  and  the 
distances  of  these  several  points  from  the  center,  C,  have  been 
recorded  in  the  field  notebook.  To  each  section  annex  the 

fraction  —  and  write  plus  and  minus  signs  on  each  side  of  each 
b 

numerator,  thus: 


q_   +C3  -    +  C2- 

b         D3  D2 


C0+   ^ 

0 


RULE  1. — To  find  the  area  (1)  multiply  each  denominator 
by  the  algebraic  sum  of  its  adjacent  numerators,  using  the  sign 
facing  the  denominator;  and  (2)  divide  the  algebraic  sum  of  the 
resulting  products  by  2. 


or, 


IT  b(C3+  c2)  +  D3C2+  D2(d-  C,)  +-] 
2  L  D!(CO  -  C2)  +  di(C0  -  c2)+  d2Cl  J 


(Let  the  student  prove  this  equation.) 

After  the  student  becomes  familiar  with  this  formula,  it  will 
not  be  necessarjr  for  him  to  actually  write  down  the  plus  and 
minus  signs  in  the  notes. 


82 


TEXT   BOOK   ON   RAILROAD    SURVEYING 


Eq.  (40)  can  be  used  for  either  cut  or  fill,  and  for  level,  three- 
level,  five-level,  and  irregular  sections. 

104.  Side-hill  Sections.  When  a  section  is  partly  in  cut  and 
partly  in  fill  as  shown  in  Fig.  38,  the  field  work  and  the  com- 
putation of  the  area  are  slightly  modified. 


\\Slope  Stake 


Slope  Stake3 


FIG.  38. 


In  this  case,  after  the  fill  at  the  center  has  been  determined, 
the  grade  point,  A,  is  found  by  trial.  Then  all  horizontal  dis- 
tances are  measured  from  A.  The  notes  would  be  recorded  as 
follows : 


_0_ 
x 


+Ci  -      +  0  +     - 


c2 
d2 


Di              0              di 
< Cut  -  X Fill  - > 

Notice  that  x  and  y  are  used  as  the  denominators  of  the  an- 
nexed fractions  instead  of  b  as  in  the  former  case. 

The  areas  of  the  cut  and  fill  are  computed  separately,  and  are 
recorded  separately  in  the  notes  (see  Fig.  34). 

105.  Uncompleted  Sections.  It  is  sometimes  necessary  to 
find  the  area  of  an  uncompleted  section — for  example,  for 
monthly  estimates — or  of  an  irregular-shaped  borrow  pit.  The 
field  work  for  this  does  not  require  that  elevations  be  carried 


EARTHWORK 


83 


from  station  to  station,  but  at  each  set-up  the  plane  of  the  horizon- 
tal line  of  sight  is  taken  as  the  reference  plane.  For  example, 
in  Fig.  39,  A  B  is  the  reference  plane  whose  elevation  is  not 


FIG.  39. 


necessary  for  the  determination  of  the  area  of  the  section.     The 
notes  should  be  written  thus: 

+  c4  -     +C3-     +  C2^-     +Ci  -     +  C0  +     -  ci  + 


-_c2 
d2 


-  c4 


0 

-c. 

0 


d, 


Notice  carefully  the  end  fractions  annexed  in  this  case.  The  area 
is  found  by  Rule  1. 

Volumes  of  Earthwork 

Earthwork  is  paid  for  by  the  cubic  yard,  and  in  order  to 
compute  the  cost  it  is  necessary  to  find  the  volume  of  the  material 
handled.  The  total  volume  is  the  sum  of  the  -volumes  of  the 
prismoids  formed  by  adjacent  cross-sections.  The  volume  of 
the  prismoid  between  two  adjacent  cross-sections  may  be  found 
(1)  by  the  end  area  method,  and  (2)  by  the  prismoidal  method. 

106.  End  Area  Method.  In  this  method  the  prismoid  is 
treated  as  a  prism,  whose  cross-section  is  the  mean  of  the  two  end 
areas  of  the  prismoid. 

Z 


v  (cu.  yds.)  = 


27 


(41) 


where  v  is  the  volume,  Ai  and  Ai  the  end  areas  in  sq.  ft.,  and 
I  is  the  distance  between  cross-sections  in  feet. 


84  TEXT   BOOK   ON    RAILROAD    SURVEYING 

For  I   =   100  ft., 

v  =  1.85  (Ai  +  A2) (41a) 

This  method  is  exact  when  the  end  areas  are  equal,  but  is  only 
approximate  when  they  are  unequal.  The  result  is  usually  too 
large  and  the  maximum  error  occurs  when  one  end  area  is  zero, 
and  is  about  16  per  cent. 

Owing  to  its  simplicity,  and  to  the  fact  that  sections  usually  do 
not  change  rapidly,  this  method  is  almost  universally  used. 

Table  12  gives  the  cubic  yards  per  100  ft.  for  center  depths 
from  1  to  60  ft. 

Table  13  gives  the  cubic  yards  per  100  ft.  for  end  areas  from 
1  to  1000  sq.  ft. 

107.  Prismoidal  Method.  When  the  changes  from  cut  to  fill 
are  frequent  and  abrupt  or  the  work  is  in  rock,  greater  accuracy 
may  be  required;  in  this  case  the  prismoidal  method  is  used. 

From  Simpson's  rule, 

.  Ai  +  4Am  +  A2       , 

Mean  area  ol  a  pnsmoiu  =  —       — ^ —     — ,  where  A\  and 

A 2  are  the  end  areas  and  Am  the  area  of  a  section  midway  be- 
tween the  ends.  Then, 

I    Z  AI  +  4  Am  -h  A2 
V  =  27  X  "       -6~  (42) 

This  is  called  the  prismoidal  formula.  This  form  however  is  in- 
convenient to  use  since  it  requires  the  area  of  a  mid-section. 
It  is  therefore  more  convenient  to  find  the  difference  between 
the  prismoidal  and  end  area  formulas  and  to  apply  this  difference 
to  the  latter.  Subtracting  the  prismoidal  formula  from  the  end 
area  formula  and  reducing, 

Prismoidal  Correction  =  ^j  (Ai  —  2Am  +  A2) 

Putting  AI,  A2,  and  Am  in  terms  of  their  slope  stake  dimen- 
sions— remembering  that  the  dimensions  of  Am  are  the  average 
of  those  of  AI  and  A2 — and  reducing,  we  have 

Pris.  Cor.  =  (d  -  C2)  (W2  -  WO    .     .     .     (43) 


EARTHWORK 


85 


in  which  Ci  and  Cz  are  the  center  cuts  or  fills  at  AI  and  A2,  and 
Wi  and  Wz  are  the  corresponding  distances  between  slope  stakes. 

When  1=  100  ft.,  Pris.  Cor.  =  0.31  (Ci  -  C2)  (W2  -  Wi)     (43a) 

This  correction  is  added  algebraically  to  Eq.  (41).  The  result  is 
exact  for  three-level  sections  and  is  sufficiently  accurate  for 
any  other  form  of  cross-section. 

108.  Earthwork  on  Curves.  The  volume  on  curves  is  gen- 
erated by  a  section  moving  perpendicular  to  the  center  line  of  the 
curve.  The  section  considered  is  the  mid-section  of  the  solid. 
The  distance  traveled  by  this  section  is  the  distance  traveled  by 
its  center  of  gravity,  and  it  is  only  when  the  center  of  gravity 
lies  in  a  vertical  line  passing  through  the  center  line  of  the  curve 
that  the  volume  as  computed  by  the  prismoidal  method  is  exact. 
If  the  center  of  gravity  is  inside  the  center  line,  the  results  ob- 
tained by  the  prismoidal  method  are  too  large;  if  outside,  they 
are  too  small.  The  difference  between  the  true  volume  and  the 
volume  determined  by  the  prismoidal  method  is  called  the 
curvature  correction. 

If  the  volumes  are  computed  by  the  end  area  method,  it  is 
inconsistent  to  apply  the  curvature  correction ;  but  if  the  pris- 
moidal method  is  used,  the  curvature  correction  should  be  ap- 
plied, especially  in  rock  work. 

The  curvature  correction  is  computed  as  follows: 

Fig.  40  represents  the   mid-section   of   the   solid.     Construct 


FIG.  40. 


BC  with  the  same  slope  as  CF.     Then  the  center  of  gravity  of 
the  part  shown  shaded  is  in  a  vertical  lins  through  C  and  there- 


86  TEXT   BOOK    ON   RAILROAD    SURVEYING 

fore  there  is  no  correction  to  be  applied  to  this  portion.  The 
center  of  gravity  of  the  triangle  ABC  is  at  G,  which  is  the  dis- 
tance x  from  the  center,  C. 

The  volume  generated  by  the  triangle  ABC  is 

v  =  ABC  X  (R  H-x)  X  I 

in  which  /  is  the  angle  at  the  center  expressed  in  radians. 
The  volume  computed  by  the  prismoidal  method  is 

vi  =  ABC  X  R  X  I 
The  curvature  correction  then  is 

C.  C.  =  v  -  vi  =  ABC  X  x  X  I 

2        1 
But  x    =  g  X  -^  X  (Di+di).    Substituting  this  value  of  x  and 

expressing  /  in  degrees,  we  have 

C.  C.  =  0.006  ABCX  (Dx  +  di)  X 1° (44) 

For  100  ft.,  7  becomes  D,  the  degree  of  the  curve. 

Haul  and  Overhaul 

109.  Haul.  Haul  is  the  work  done  in  transporting  the  ma- 
terial from  the  excavation  to  its  place  in  the  embankment. 
Its  unit  is  the  yard-station,  or  the  work  done  in  transporting 
one  cubic  yard  100  ft.  The  term  haul  is  frequently  used  to 
indicate  either  the  yardage  or  the  distance,  but  this  must  not 
be  confused  with  the  true  meaning. 

Contract  prices  are  usually  based  on  a  maximum  length  of 
haul — generally  500  ft. — called  the  free-haul  distance.  If  the 
material  is  transported  a  longer  distance  the  extra  work  in- 
volved is  termed  overhaul  and  the  contractor  is  allowed  extra 
compensation  at  a  certain  price  per  yard-station.  The  con- 
tractor is  therefore  paid  for  all  excavation  and  for  overhaul,  but 
he  is  not  paid  for  making  the  fills  nor  for  haul  inside  the  free-haul 
distance. 

The  amount  of  excavation  is  readily  determined  from  the 
cross-section  notes,  hence  this  portion  of  the  contractor's  re- 


EARTHWORK  87 

muneration  presents  no  particular  difficulties  except  when  the 
excavation  involves  several  different  kinds  of  materials  to  be 
paid  for  at  different  prices.  The  amount  of  overhaul,  however, 
depends  on  the  plan  of  distributing  the  excavated  material, 
and  the  problem  is  to  determine  an  economical  distribution. 
This  problem  is  not  of  great  difficulty,  but  unfortunately  it  has 
often  been  neglected,  resulting  in  uneconomical  work,  misunder- 
standings with  the  contractor,  and  frequent  law-suits. 

The  limit  of  profitable  haul  is  that  distance  at  which  the  cost 
of  overhaul  equals  the  cost  of  excavating  a  yard  of  earth.  For 
example,  if  the  contract  price  is  24  cents  a  yard  for  excavation 
and  If  cents  a  yard-sta.  for  overhaul,  and  the  free-haul  distance 

24 
is  500  ft.,  the  limit  of  profitable  haul  is  p  -f  5  =  21  Sta.* 

110.  Case  1.  Fig.  41  represents  a  fill  between  two  cuts.  It 
is  readily  seen  that  there  is  more  than  enough  earth  in  the  cuts 
to  make  the  fill,  and  the  question  is  how  much  of  the  fill  shall  be 
made  from  each  cut. 

The  points  B  and  D  are  the  free-haul  distance  apart  and  are 
located  on  the  profile  so  that  the  volume  of  cut  between  B  and  C 
equals  the  volume  of  fill  between  C  and  D.  F  and  H  are  simi- 
larly located  with  respect  to  G. 

Since  the  cut  to  the  right  is  the  deeper,  it  is  evident  that  the 
fill  can  be  made  with  the  minimum  haul  if  the  greater  volume  of 
earth  is  taken  from  the  right-hand  cut.  The  problem  is  to 
locate  A,  E,  and  «/,  which  are  the  limits  of  the  distribution  from 
each  of  the  two  cuts.  This  can  not  be  done  by  inspection  nor 


*  A  yard  of  earth  excavated,  hauled  700  ft.,  and  dumped  into  the  fill 
costs  the  railroad  company  24  cts.  plus  3  cts.  overhaul.  If  the  yard 
excavated  had  been  hauled  only  a  few  feet  to  one  side  of  the  cut  and 
dumped,  and  another  yard  of  earth  had  been  taken  from  a  borrow 
pit  along  side  of  the  fill  and  placed  hi  the  fill,  the  cost  would  have 
been  48  cts.  It  is  evident  then  that  it  is  much  cheaper  for  the  railroad 
to  pay  the  additional  amount  for  overhaul. 

In  the  above  example,  the  saving  was  21  cts.  If  the  haul  had  been 
longer,  the  saving  would  have  been  less.  But  the  railroad  saves  money 
up  to  the  point  where  the  cost  of  overhaul  equals  the  cost  of  making  a  yard 
of  fill  from  a  borrow  pit.  Beyond  this  point  there  is  an  increasing  loss, 
and  when  twice  the  limit  of  profitable  haul  is  reached,  this  loss  equals 
the  previous  saving.  The  limit  of  profitable  haul  is  considered  as  that 
distance  at  which  the  profit  is  the  largest. 


88 


TEXT   BOOK    ON   RAILROAD    SURVEYING 


can  it  be  accomplished  easily  from  the  cross-section  notes.  It 
can  be  done  most  easily  and  accurately  by  constructing  a  dis- 
tribution diagram. 

The  first   step   in  making    the    distribution    diagram    is   to 
construct  the  mass  curve,  which  is  done  in  the  following  manner : 


FIG.  41. 

Choose  a  horizontal  base  line  near  the  bottom  of  the  profile  sheet, 
and  project  the  grade  points  C  and  G  onto  it  at  C"  and  Gf  as  shown 
in  Fig.  41. 

Then  beginning  at  C'  plat  to  the  chosen  scale — usually  1,000 
cu.  yds.  to  the  inch — a  curve  such  that  the  ordinate  to  it  at  any 
point  is  equal  to  the  total  volume  of  earthwork  between  that 
point  and  C".  From  G'  plat  a  similar  curve  to  the  same  scale. 
The  ordinates  to  these  curves  are  computed  at  the  regular 
stations  from  the  cross-section  notes.  Allowance  for  shrinkage 
or  growth  must  be  made  by  increasing  the  fill  volumes  by  the 
amount  of  shrinkage  or  decreasing  them  by  the  amount  of 
growth. 

The  second  step  is  to  determine  the  limits  of  free  haul.  This 
is  done  by  locating  a  horizontal  line  equal  in  length  to  the  free- 
haul  distance  in  such  a  position  that  its  ends  are  in  the  mass 
curve  as  at  B'D'  and  F' H'.  These  points  are  projected  up  to 
locate  B,  D,  F,  and  H.  The  ordinate  to  B'D'  is  the  free-haul 
yardage  past  C,  and  the  ordinate  to  F'H'  is  the  free-haul  yardage 
past  G. 

The  third  step  is  to  find  the  positions  of  A,  E,  and  J  so  that  the 
haul  will  be  a  minimum.  For  all  practical  purposes  this  occurs 


EARTHWORK  »9 

when  A  E  =  EJ.  Therefore  in  the  distribution  diagram  find 
by  trial  the  positions  of  the  two  horizontal  lines  A'E'  and  E"Jr 
of  equal  length,  with  Er  and  E"  on  the  same  vertical  line.  These 
points  are  projected  onto  the  profile  to  locate  A,  E,  and  /. 
These  points  should  be  located  in  the  field  and  distinctly  marked 
for  the  guidance  of  the  contractor. 

The  ordinates  between  B'D'  and  A'E'  and  also  between  F'H' 
and  E"J'  give  the  total  volumes  on  which  there  is  an  overhaul 
charge.  The  total  volume  moved  is  given  by  the  ordinates  from 
the  base  line  to  A'E'  and  E"J'. 

Since  the  mass  curve  is  a  curve  between  distance  and  total 
volumes,  it  follows  that  the  area  inside  the  curve  represents 
haul  in  yard-stations.  Consequently  the  overhaul  is  given  by 
the  shaded  portion  between  the  curves  and  the  lines  A'E' 
and  E"J'. 

If  the  ordinate  B  'b  is  bisected — dividing  the  overhaul  yardage 
into  two  equal  parts — and  the  horizontal  line  M  N  is  drawn,  the 
two  points  M  and  N  will  be  directly  under  the  centers  of  fnass  of 
the  cut  and  the  fill.  Therefore  M  N  minus  the  free-haul  distance 
is  the  overhaul  distance  corresponding  to  B  '&;  and  the  overhaul 
can  be  found  by  multiplying  B'b  by  this  distance.  This  method 
is  frequently  easier  of  application  than  that  of  determining  the 
areas  direct.  The  overhaul  past  G  can  be  determined  similarly. 


C' 

1 
3 

FIG.  42. 

G' 

111.  Case  2.  In  Fig.  42,  C  and  G  are  so  far  apart  and  the  cuts 
are  so  long  that  it  is  evident  that  the  limit  of  profitable  haul  will 
be  reached.  Construct  the  mass  curve  and  locate  the  free-haul 


90  TEXT   BOOK   ON   RAILROAD   SURVEYING 

limits  as  before.  Lay  of  A'E'  and  K'J'  each  equal  to  the  limit 
of  profitable  haul.  The  centers  of  mass,  M,  N,  P,  and  Q,  are 
found  as  in  Case  1.  The  points  A',  E',  Kf,  and  /'  projected 
onto  the  profile  give  the  points  A,  #,  K,  and  J \  which  should 
be  marked  in  the  field. 

The  fill  between  E  and  K  must  be  borrowed  and  the  yardage 
required  is  given  by  the  difference  of  the  ordinates  to  Er  and  K" . 
The  amount  of  overhaul  is  determined  as  in  the  former  case. 
The  complement  of  this  case  is  the  determination  of  waste  in  a 
long  cut. 

112.  Case  3.  In  case  earth  can  not  be  moved  past  some  point, 
the  distribution  diagram  is  constructed  as  before  except  that 
the  horizontal  distribution  lines  (A'E'  and  E"J'  in  Fig.  41) 
must  start  at  the  fixed  point.  The  remainder  of  the  work  is  done 
as  before.  This  condition  frequently  occurs— usually  at  stream 
crossings, 


CHAPTER  VI 

TURNOUTS,  CONNECTIONS,  AND  CROSSINGS 
113.  The  subjects  treated  in  this  chapter  are  as  follows. 

,   /  Switches 
1.  Turnouts  <  ^ 

(  Frogs 


2.  Connections 


Diverging      /  From  Tangents 

Tracks       \  From  Curves 

(  From 

c,.,.                       Tangents 
Sidings           {  ^ 
1  From 

Parallel 

Curves 

Tracks 

From 

Crossovers 

Tangents 
From 

Curves 

Intersecting    (Tangents 

Tracks        \  Curves 

3.  Crossings 


114.  Definitions. 


1.  Turnouts 


A  switch  is  a  device  to  deflect  at  will  the  wheels  of  a  train 
from  the  track  on  which  they  are  running. 

A  switch  stand  is  a  device  for  operating  a  switch  (see  Fig.  45). 

A  frog  is  a  device  at  the  intersection  of  two  rails  to  permit 
the  passage  of  the  wheel  flanges. 

A  guard  rail  is  a  short  section  of  rail  placed  opposite  the 
frog  to  guide  the  wheels  so  that  the  flanges  will  pass  through 
the  frog  properly. 

A  turnout  is  a  combination  of  a  switch,  a  frog,  and  the  necessary 
connecting. rails,  etc.,  to  permit  rolling  stock  to  pass  from  one 
track  to  another  (see  Fig.  45). 

A  siding  is  a  secondary  track  parallel  to  the  main  track  which 
is  used  for  the  storage  of  rolling  stock  or  for  the  passing  of  trains. 

91 


92  TEXT   BOOK   ON   RAILROAD    SURVEYING 

It  is  connected  to  the  main  track  by  a  turnout  at  one  or  both 
ends. 

A  crossover  is  a  connection  at  an  intermediate  point  between 
two  parallel  adjacent  tracks  (see  Fig.  45). 

115.  Switches.  Switches  are  of  three  types,  viz.,  Split 
Switches,  Tongue  Switches,  and  Stub  Switches. 

The  Split  Switch,  Fig.  43a,  is  the  form  universally  used  on 
steam  roads.  It  consists  of  two  wedge-shaped  sections  of  rails 
connected  by  tie  rods  and  sliding  on  metal  plates  placed  on  the 
ties.  The  broad  end  or  heel  of  each  point  rail  acts  as  a  pivot  by 
being  fastened  to  the  lead  rails  by  means  of  the  usual  splice  bars. 
The  point  moves  through  a  distance  of  about  5J  inches,  which  is 
called  the  throw  (t),  and  is  controlled  by  a  switch  stand  placed 
outside  the  track  on  a  long  tie  (sometimes  two)  called  a  head- 
block.  The  distance  between  the  gage  lines  at  the  heel  is  called 
the  heel  spread  (h).  The  angle  between  the  gage  lines  of  the 
switch  and  the  main  rail  is  called  the  switch  angle  (s).  The 
width  of  the  point  is  usually  \"  (when  new).  From  Fig.  43a 
it  is  seen  that 

sin  s  =  *~         ......     (45) 


where  p  is  the  width  of  the  point,  and  I  the  length  of  the  switch 
rail.  Split  switches  vary  in  length  from  10  to  33  feet,  depending 
on  the  flatness  of  the  turnout  desired.  The  point  rails  are  cut 
from  rails  of  standard  length  so  as  to  have  no  waste.  For  30-ft. 
rails,  10,  12,  15,  18,  20,  and  30-ft.  switch  points  are  used,  and 
for  33-ft.  rails,  11,  13,  16J,  20,  22,  and  33-ft.  The  most  common 
lengths  are  15  and  16J  ft. 

The  Tongue  Switch,  Fig.  43b,  consists  of  a  steel  wedge  or 
tongue  pivoted  at  one  end  and  moving  in  a  heavy  cast-iron 
frame  so  arranged  that  pavement  may  be  built  around  it.  This 
is  the  common  form  on  street  railways,  where  it  is  usually  used 
singly  with  a  rigid  fitting  called  a  mate  in  the  opposite  rail. 
Steam  roads  use  them  only  for  turnouts  in  pavement,  and  then 
usually  in  pairs,  where  the  action  is  the  same  as  in  a  split  switch. 

A  Stub  Switch,  Fig.  43c,  consists  of  a  pair  of  ordinary  rails 
fastened  together  with  tie  rods.  The  rails  are  spiked  for  part  of 
their  length,  the  remainder  being  thrown  to  match  the  stub  ends 
of  the  lead  rails.  A  stub  switch  differs  from  a  split  switch  in  that 
the  toe  is  fixed  and  the  heel  moves.  This  form  of  switch  may  be 


TURNOUTS,    CONNECTIONS,   AND   CROSSINGS  93 


rmnnnnnn 


HjS.fi        [fil  J!L  15L__  J5L  ALJ3LL    _U3,  - 


LI  U  U  U  LJ 

FIG.  43a. 


MATE 


SWITCH 
FIG.  43b. 


FIG.  43c. 


94 


TEXT   BOOK   ON    RAILROAD    SURVEYING 


employed  to  advantage  on  construction  and  mine  tracks  owing 
to  its  cheapness  and  ease  of  installation.  A  stub  switch  should 
never  be  used  on  a  standard  road. 

116.  Frogs.  Frogs  are  of  two  types,  viz.,  Rigid  Frogs  and 
Spring-rail  Frogs.  Frogs  of  either  type  are  universally  made 
with  straight  gage  lines  (except  on  street  railways) .  The  point 
rails  are  finished  with  a  blunt  point  usually  \"  wide.  The  dis- 
tance P  between  this  actual  and  the  theoretical  point  or 
intersection  of  gage  lines  equals  the  width  multiplied  by  the 
frog  number. 

Fig.  44&  shows  the  common  form  of  rigid  frog  and  the  names 
of  the  various  parts.  The  essential  feature  of  the  rigid  frog 
is  that  both  flangeways  are  open  and  must  be  jumped  by  passing 
wheel  treads. 


Theoretical  Point  - 


^Actual  Point 


Flange  way 


Short  Point  Rail 


Wing  RaiK 

FIG.  44a.    Rigid  Frog. 


Main  Point  Rail 


FIG.  44b.    Spring-rail  Frog. 


Fig.  44b  shows  the  common  form  of  a  Spring-rail  Frog.  In  this 
type  one  wing  rail  (sometimes  both)  is  made  movable.  It  is 
normally  held  against  the  point  rails  by  springs.  A  wheel  on 
the  main  track  therefore  has  an  open  flangeway  and  a  con- 
tinuous support  for  the  tread,  while  a  wheel  on  the  turnout  must 
spring  the  wing-rail  open  to  allow  the  flange  to  pass,  and  at  the 
same  time  the  treads  lack  continuous  support  as  in  a  rigid  frog. 
Guard  rails  are  absolutely  essential  to  the  safe  operation  of  a  spring- 


FIG.  45.    Crossover. 


96  TEXT   BOOK   ON    RAILROAD   SURVEYING 

rail  frog.     Spring-rail  frogs  are  used  on  main  track  turnouts,  and 
rigid  frogs  in  yards. 

A  special  type  of  frog  called  the  movable  point  frog  is  used  in 
connection  with  crossings  particularly  with  "  slip  switches." 

117.  Frog  Angle,  F.     The  frog  angle   is   the   angle   between 
the  gage  lines  at  their  point  of  intersection.     Obviously  the  frog 
angle  may  have  any  value,  but  only  values  of  about  2°  to  8°  are 
practical  for  turnouts.     For  turnouts  a  few  standard  values  of 
the  frog  angle  can  be  chosen  which  will  meet  all  the  practical 
requirements.     These  angles  are  chosen  to  agree  with  simple 
values  of  the  Frog  Number  which  is  a  more  convenient  method  of 
designating  frogs  than  by  their  angles. 

118.  Frog  Number,  N.     The  Frog  Number  is  the  ratio  of  the 
a  .ial  length  to  the  spread,  i.e.,  it  is  the  distance  measured  along 
the  bisector  of  the  angle  in  which  the  gage  lines  diverge  a  unit 
distance.     Reducing  this  ratio  to  a  trigonometric  function  we 
have, 

N  =  1  cot  JF  (46) 

Values  of  N  from  7  to  15  are  commonly  used,  but  for  special 
purposes  values  from  3  to  27  are  occasionally  used. 

119.  Location  of  Turnouts.     The  first  step  in  staking  out  a 
turnout  (whether  on  straight  or  curved  track)  is  to  locate  the 
frog.     The  heul  or  toe  of  the  frog  should  come,  if  possible,  at  a 
regular  rail  joint  in  order  to  avoid  short  pieces  of  rail  in  the 
track.     This  fixes  the  position  of  the  P.F.  (point  of  frog).     The 
next  step  is  to  locate  the  P.S.  (point  of  switch),  which  is  done  by 
measuring   along   the   main   track   the  distance   L    (the  lead). 
The  next  step  is  to  locate  the  outside  rail  of  the  turnout  curve. 
This  is  done  by  offsets  from  the  outside  main  rail  at  the  middle 
and  quarter  points  of  the  turnout  curve.     The  inner  turnout 
rail  is  set  by  gage  from  the  outer. 

120.  Turnouts    from    Straight   Track.     Since    the   frogs    and 
switches  are  straight,  the  alinement  of  a  turnout  is  not  a  simple 
curve  from  a  tangent,  but  consists  of  two  short  pieces  of  tangent 
making  a  fixed  angle  with  the  main  track  rails  connected  by  a 
curve. 

In  triangle  1,  Fig.  46,  the  hypotenuse  and  angles  are  known. 
Solve  for  m  and  n.     In  triangle  2,  q  =  g  —  h  —  n  and  the  angles 


TURNOUTS,    CONNECTIONS,    AND    CROSSINGS  97 


are  known.  Solve  for  j  and  c.  c  is  the  chord  of  the  outside  lead 
rail  and  the  central  angle  (F—s)  is  known,  hence  the  radius  of 
the  outside  rail  is 

-        .....     (47) 


The  distance  along  the  straight  track  from  the  P.P.  to  the 
P.S.  is  called  the  lead,  L.     From  the  figure 

L  =  l+j+m  +  P       .....     (48) 


P.S. 


h)        (Vi  I 


'=N-x   Width  of  frog  point 


FIG.  46. 


To  compute  the  offsets  for  the  lead  rail,  consider  its  gage  line 
produced  back  to  a  tangent  parallel  to  the  main  rail.  The 
distance  between  these  tangents  is 

e  =  h  —  (R  +  ig)  vers  s 

The  offset  from  the  main  rail  to  the  quarter  point  of  the  lead  rail 
is  then, 

yi  =  e  +  (R  +  ig)  vers  [s  +  J(F  -  s)]     (Slide  Rule) 
For  the  middle  point, 

y2  =  e  +  (R  +  ig)  vers  [s  +  J(F  -  s)]        "          "         (49) 
For  the  three-quarters  point, 

y3  =  e  +  (R  +  ig)  vers  [s  +  f  (F  -  s)]        "      "  '^ 

121.  Turnouts  from  Curved  Track.  It  can  be  shown  (the 
solution  is  too  involved  to  be  included  here)  that  the  lead  of  a 
turnout  from  a  curved  track  is  almost  identical  with  the  lead 
of  a  turnout  from  straight  track  with  the  same  frog  number. 
And  further,  that  the  degree  of  curve  of  the  lead  rails  is  equal  to 
the  degree  of  curve  of  the  turnout  from  straight  track  increased 


98  TEXT   BOOK   ON   RAILROAD    SURVEYING 

or  diminished  by  the  degree  of  the  main  curve  depending  on 
whether  the  turnout  is  on  the  inside  or  the  outside  of  the  main 
curve. 

Therefore  a  turnout  on  a  curve  is  staked  out  with  the  same  dimen- 
sions as  one  with  the  same  frog  number  on  straight  track. 

122.  Double  Turnouts.     Occasionally    two    turnouts    to    op- 
posite sides  of  the  main  track  are  located  at  the  same  point. 
This  involves  the  use  of  two  regular  frogs  in  the  main  rails,  a 
third  or  crotch  frog  at  the  intersection  of  the  lead  rails,  and  a 
three-throw   switch.     Such   a   switch   is   structurally   weak   and 
should  never  be  used  in  a  main  track.     And  further,  the  condi- 
tions under  which  such  arrangements  are  absolutely  necessary 
are  extremely  rare.     It  is  better  practice  to  use  two  separate 
turnouts,  placing  one  switch  ahead  of  the  other  and  if  necessary 
using  a  sharper  turnout  to  save  distance,  than  to  use  a  three-throw 
switch. 

Double  turnouts  will  therefore  not  be  further  considered. 

123.  Practical  Leads.     Since  it  is  undesirable  to  use  short 
pieces  or  odd  lengths  of  rails,  it  is  the  universal  practice  to 
modify  the  theoretical  leads  as  computed  above  (Eq.  48),  so  as 
to  use  convenient  lengths  of  lead  rails.     The  difference  between 
the  theoretical  leads  and  the  practical  leads  is  never  very  great, 
and  the  turnout  is  located  as  above  except  for  this  modification 
in  the  lead.     This  has  the  effect  of  changing  the  turnout  curve 
from  a  true  circle,  but  the  amount  is  inappreciable.     There  is  also 
a  difference  in  the  length  of  the  straight  and  the  curved  lead  rails. 
The  best  practice  is  to  correct  this  by  making  the  toe  length  of 
the  frog  on  the  turnout  side  longer  than  on  the  main-track  side. 

Every,  railroad  has  its  standard  turnouts,  all  dimensions  of 
which  are  worked  out  and  tabulated.  The  turnouts  are  then 
installed  by  the  trackmen,  the  only  duty  required  of  the  en- 
gineer being  to  locate  the  P.P.  and  occasionally  the  P.S.  The 
engineer  therefore  has  little  occasion  to  use  the  functions  of  the 
turnout  proper.  His  problem  is  to  connect  the  turnout  to  the 
required  track. 

Table  2  gives  the  practical  switch  leads  recommended  by  the 
American  Railway  Engineering  Association;  also  the  rectangular 
co-ordinates  to  the  quarter  and  center  points  on  the  gage  side 
of  the  curved  rail,  referred  to  the  point  of  switch  as  origin. 


TURNOUTS,    CONNECTIONS,   AND    CROSSINGS 


99 


TABLE  2 

Practical  Leads  as  Recommended  by  the  American  Railway 
Engineering  Association 


8^ 

1 

0.1 

|| 

Rectangular     Co-ordinates     to     the 
8uarter     and     Center     Points     on 

I? 

y 

p 

age    Side    of    Curved    Rail,    Re- 

<H 2 

* 

ll 

!•§ 

ferred  to  Point  of  Switch  as  Origin 

"Sal 

£ 

gl 

°3 

3J!s 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

7 

362  .  08 

15°  52' 

26.72 

36.93 

47.11 

0.97 

1.71 

2.74 

62.10 

8 

487  .  48 

11     46 

28.37 

39.91 

51.45 

.02 

1.78 

2.91 

67.98 

9 

605  .  18 

9     29 

28.75 

40.98 

53.19 

.02 

.76 

2.75 

72.28 

10 

790  .  25 

7     15 

30.28 

44.05 

57.81 

.06 

.84 

2.85 

77.93 

11 

922  .  65 

6     13 

40.74 

56.47 

72.19 

.08 

.84 

2.87 

94.31 

1.3 

1098  .  73 

5     13 

43.99 

60.65 

77.28 

.15 

.90 

2.91 

100.90 

15 

1744.38 

3     17 

55.49 

77.95 

100.41 

.01 

.78 

2.85 

133  .  28 

IS 

2546.31 

2     15 

58.73 

84.46 

110.10 

.04 

.82 

2.86 

146.51 

20 

3257  .  26 

1     46 

61.84 

90.21 

118.50 

.08 

.88 

2.93 

157.42 

24 

4886.16 

1     10 

67.82 

100.21 

132.59 

.27 

.97 

3.00 

177.22 

2.  Connections 
DIVERGING  TRACKS. 
124.  Case  1.    From  Straight  Track. 


FIG.  47. 
Given: 
7    =  the  intersection  angle  between  a  diverging  track  and  a 

straight  main  track. 

N  =  the  frog  number  chosen  for  the  turnout. 
R   =  the  chosen  radius  of  the  connecting  curve. 


100  TEXT   BOOK   ON   RAILROAD   SURVEYING 

K    =  the   frog   tangent,   i.e.,   the   distance   from   the   actual 
P.P.  to  the  T.C.  of  the  connecting  curve. 

P    =  the  distance  between  the  theoretical  and  the  actual  point 

of  frog. 
It  is  required  to  locate  the  P.P.,  the  T.C.,  and  the  C.T. 

In  triangle  1,  Fig.  47,  all  the  angles  are  known  and  the  side 
m  =  gN  +  P  +  K  +  R  tan  i(/  —  F),  whence  the  sides  n  and 
j  can  be  computed. 

The  point  B  and  the  C.  T.  of  the  connecting  curve  are  located 
from  A  by  the  distances  n  andj  +  R  tan  J(7  —  F),  respectively. 
The  P.P.  is  located  from  B  by  the  co-ordinates  g  N  +  P  and  |gr 
=  2.35.  The  T.C.  of  the  connecting  curve  is  located  from  B 
by  the  angle  F  and  the  distance  gN  +  P  +  -K- 

125.  Case  2.    From  Curved  Track. 

(a)  Turnout  from  the  inside  of  a  curved  main  track. 

The  data  and  requirements  of  this  problem  are  the  same 
as  in  the  preceding  problem  except  that  the  main  track  is  on  a 
D°  curve. 

In  triangle  1,  Fig.  48,  the  hypotenuse  and  the  angles  are  known 
and  the  other  two  sides  are  computed  (as  shown),  whence, 

In  triangle  2,  the  base  and  altitude  are  found  by  arithmetic, 
and  the  triangle  is  solved  for  the  hypotenuse  and  the  angle  a. 

In  triangle  3,  the  hypotenuse  and  angles  are  known  and  the 
other  two  sides  are  computed  (as  shown),  whence, 

In  triangle  4,  one  side  and  the  hypotenuse  may  be  found  by 
arithmetic,  and  the  triangle  solved  for  the  third  side  and  the 
angle  b.  Then, 

x  =  a  +  b  +  (90°  -  I)  -  (90°  -  F) 

=  a  +  b-I  +  F        (50) 

The  point  H  opposite  the  theoretical  frog  point  is  located  from 
A  by  measuring  along  the  center  line  the  distance  A  H  (in  feet) 

=  -^  100.     The  C.T.  is  located  from  A'by  the  distance  AE  = 

AG  -  EG.  (EG  =  JO.)  The  T.C.  is  best  located  as  follows:  set- 
up at  H;  back-sight  on  A',  turn  off  \x  to  get  on  tangent;  locate 
M  for  a  temporary  back-sight;  plunge  telescope  and  set  point 


TURNOUTS,    CONNECTIONS,    AND    CROSSINGS         101 


FIG.  48. 


FIG.  49. 


102 


TEXT   BOOK   ON   RAILROAD    SURVEYING 


B  a  distance  gN  from  H]  set-up  at  B  and  back-sight  *  on  M; 
turn  off  frog  angle  and  locate  the  T.C.  on  this  line  a  distance 
gN  +  P  +  K  from  B]  set  point  Q  for  a  temporary  back-sight. 
Set-up  at  the  T.C.,  back-sight  on  Q,  and  run  in  the  connecting 
curve. 

(6)  Same  problem  as  (a)  except  that  the  main  line  curve  ends 
between H  and  A. 

In  Fig.  49,  A  is  the  intersection  of  the  diverging  track  tangent 
and  the  main  line  tangent.  A'  is  the  C.T.  of  the  main  track 
curve.  The  distance  A  A'  is  measured  in  the  field.  Then  the 
angle  x,  the  distance,  HA'  and  A'E'  are  determined  as  in  the 
preceding  problem. 

In  triangle  5,  the  side  A  A'  and  the  angles  are  known,  hence 
I  can  be  computed. 

The  C.T.  (at  E)  is  located  from  A  by  the  distance  A'E'  -  I. 

(c)  Turnout  from  the  outside  of  a  curved  main  track. 


R  ein  I 


FIG.  50. 


*A  back-sight  is  a  sight  taken  on  a  point  of  known  position  for  the 
purpose  of  orienting  the  instrument,  and  may  be  either  back  of  or 
in  front  of  the  instrument. 


TURNOUTS,    CONNECTIONS,   AND    CROSSINGS        103 

In  triangle  1,  Fig.  50,  the  hypotenuse  and  angles  are  known, 
and  the  other  two  sides  are  computed  (as  shown),  whence, 

In  triangle  2,  the  base  and  altitude  are  found  by  arithmetic. 
The  triangle  is  solved  for  the  hypotenuse  and  the  angle  a. 

In  triangle  3,  the  hypotenuse  and  angles  are  known  and  the 
other  two  sides  are  computed  (as  shown),  whence, 

In  triangle  4,  one  si^le  and  the  hypotenuse  can  be  found  by 
arithmetic.  The  triangle  is  solved  for  the  third  side  and  the 
angle  b.  Then 

x  =  b  -  (90°  -  I)  -  [a  -  (90°  -  F)] 

=  b  +  I-a-F (51) 

The  field  work  is  identical  with  that  of  Problem  (a). 

(d)  Same  problem  as  (c)  except  that  the  main  track  curve  ends 
at  A'  between  A  and  H. 


FIG.  51. 


This  problem,  Fig.  51,  bears  the  same  relation  to  (c)  that  (6) 
bears  to  (a). 

If  /  is  greater  than  90°  in  any  of  these  four  problems,  the 


104 


TEXT   BOOK    ON    RAILROAD    SURVEYING 


various  lines  used  in  the  solution  will  have  relatively  different 
positions,  but  the  general  order  of  solution  is  the  same,  and  if 
followed  out  no  difficulty  should  arise. 

PARALLEL  TRACKS.    SIDINGS. 
126.  Case  1.     Straight  Tracks. 

(a)   Tangent  and  simple  curve. 

The  best  way  to  connect  a  turnout  to  a  parallel  siding  is  to 
make  the  frog  tangent  of  such  a  length  that  it  can  be  connected 
to  the  siding  by  a  simple  curve  of  about  the  same  radius  as  that 
of  the  lead  rails.  A  connecting  track  of  this  kind  is  most  easily 
located  by  co-ordinates  from  the  actual  point  of  frog. 


FIG.  52. 

In  Fig.  52,  xi,  x2,  and  £3  are  points  on  the  main  track  op- 
posite the  T.C.,  the  middle  point  of  the  curve,  and  the  C.77., 
respectively,  and  y\,  7/2,  and  ys  are  the  corresponding  offsets 
from  the  center  line  of  the  main  track.  From  the  figure, 

yi  =  p  —  R  vers  F     and  xi  =  yi  cot  F  —  gN—  P 

y2  =  p  -  R  vers  JF     "     x2  =  Xi  -f  R  (sin  F  -  sin  JF)  >  (52) 

y3  =  p  "     x3  =  xi  +  R  sin  F 


Table  3  gives  the  values  of  x  and  y  for  track  centers  from 
12  to  16  ft.  and  for  frog  numbers  from  7  to  15. 

(b)  Sometimes  the  connecting  curve  is  commenced  at  the 
heel  of  the  frog  or  a  fixed  distance  beyond  it.  In  this  case  x\ 


TABLE  3  105 

Co-ordinates  for  Locating  Parallel  Sidings  with  Tape 


fc 

£ 
§ 

§3 

fc 

0 

§£ 

§ 

H 

% 

O  II 

Xl 

X2 

X3 

yi 

y2 

|| 

0 

Variation 

1 

EH 

a° 

2 
£ 

7 

12.0 

23.31 

50.45 

77.73 

8.12 

11.03 

15°  00' 

8°     10' 

&&>& 

12.5 

26.79 

53.93 

81.21 

8.62 

11.53 

13.0 

30.28 

57.42 

84.70 

9.12 

12.03 

_«  2*Ji 

13.5 

33.76 

60.90 

88.18 

9.62 

12.53 

O  OO 

14.0 

37.24 

64.38 

91.66 

10.12 

13.03 

v 

14.5 

40.73 

67.87 

95.15 

10.62 

13.53 

iH  X  >> 

15.0 

44.21 

71.35 

98.63 

11.12 

14.03 

O      .^« 

15.5 

47.70 

74.84 

102.12 

11.62 

14.53 

t,  ao  a 

16.0 

51.18 

78  .  32  105  .  60 

12.12 

15.03 

fe.So8 

8 

12.0 

25.33 

57.73 

90.26 

7.94 

10.98 

11°  00' 

7°    09' 

0>®  35 

12.5 

29.32 

61.72 

94.25 

8.44 

11.48 

13.0 

33.31 

65.71 

98.24 

8.94 

11.98 

13.5 

37.29 

69.69 

102  .  22 

9.44 

12.48 

€•§•9 

14,  .0 

41.28 

73.68 

106.21 

9.94 

12.98 

v    X  >» 

14.5 

45.26 

77.66 

110.19 

10.44 

13.48 

T~! 

15.0 

49.25 

81.65 

114.18 

10.94 

13.98 

®      -v  '      . 

15.5 

53.24 

85.64 

118.17 

11.44 

14.48 

f_  ao  a 

16.0 

57.22 

89.62 

122.15 

11.94 

14.98 

fe.So£ 

9 

12.0 

29.59 

64.87 

100.26 

8.07 

11.02 

9°  00' 

6°    22' 

©35© 

12.5 

34.07 

69.35 

104  .  74 

8.57 

11.52 

sf  d  9 

13.0 

38.55 

73.83 

109  .  22 

9.07 

12.02 

_rt  Ji  J3 

13.5 

43.03 

78.31 

113.70 

9.57 

12.52 

yoy 

14.0 

47.51 

82.79 

118.18 

10.07 

13.02 

v 

14.5 

51.99 

87.27 

122  .  66 

10.57 

13.52 

r-J  X  r* 

15.0 

56.47 

91.75 

127.14 

11.07 

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O    -^'    . 

15.5 

60.95 

96.23 

131.62 

11.57 

14.52 

fc       § 

16.0 

65.43 

100.71 

136.10 

12.07 

15.02 

fc.S6rf 

10 

12.0 

31.23 

72.08 

113.04 

7.90 

10.98 

7°  00'  5°    44' 

&3*a 

12.5 

36.21 

77.06 

118.02 

8.40 

11.48 

13.0 

41.19 

82.04 

123  .  00 

8.90 

11.98 

J*li 

13.5 

46.17 

87.02 

127.98 

9.40 

12.48 

14.0 

51.15 

92.00 

132.96 

9.90 

12.98 

^       l~l  fl 

14.5 
15.0 

56.13 
61.11 

96.98 
101.96 

137.94 
142  .  92 

10.40 
10.90 

13.48 
13.98 

Sail 

15.5 

66.09 

106.94 

147  .  90 

11.40 

14.48 

o'  _«  3 

16.0 

71.07 

111.92 

152  .  88 

11.90 

14.98 

fc.So£ 

12 

12.0 

39.38 

86.96 

134  .  63 

8.03 

11.00 

5°  00' 

4°    46' 

°3  *a 

12.5 

45.38 

92.96 

140  .  63 

8.53 

11.50 

C3  5    •  «J 

13.0 

51.38 

98.96 

146.63 

9.03 

12.00 

c3      Q  «3 

13.5 

57.37 

104  .  95 

152  .  62 

9.53 

12.50 

"§      N  ® 

14.0 

63.37 

110.95 

158  .  62 

10.03 

13.00 

•^           r"1  CH 

14.5 

69.37 

116.95 

164.62 

10.53 

13.50 

*"•       ®  J3 

15.0 

75.36 

122  .  94 

170.61 

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®  ft  r?  W 

15.5 

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128  .  94 

176.61 

11.53 

14.50 

0         ^^ 

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134  .  93 

182  .  60 

12.03 

15.00 

Sail 

15 

12.0 

48.38 

111.08 

173.85 

7.87 

10.97 

3°  00' 

3°    46' 

®3*a 

12.5 

55.97 

118.67 

181.44 

8.37 

11.47 

jn'^J     .  M 

13.0 

63.57 

126.27 

189  .  04 

8.87 

11.97 

CS        £Q  C3 

13.5 

71.16 

133.86 

196.63 

9.37 

12.47 

w^^S, 

14.0 

78.76 

141.46 

204  .  23 

9.87 

12.97 

X            fl 

14.5 

86.35 

149  .  05 

211.82 

10.37 

13.47 

'"1             ®    ?H 

15.0 

93.95 

156.65 

219.42 

10.87 

13.97 

o  agw 

15.5 

101.54 

164.24 

227.01 

11.37 

14.47 

o    ^3 

16.0 

109.14 

171.84 

234.61 

11.87 

14.97 

fe.Su'S 

106 


TEXT   BOOK   ON   RAILROAD   SURVEYING 


and  2/1  are  known  and  the  radius  of  the  connecting  curve  must 
be  computed.  The  other  dimensions  are  then  computed  as 
before.  Although  this  method  flattens  the  connecting  curve,  it 
increases  the  distance  from  the  P.P.  to  the  T.C.  There  is 
nothing  gained  by  making  the  connecting  curve  flatter  than 
that  of  the  turnout  itself,  and  also  the  greater  length  of  tangent 
is  advantageous. 

(c)  Sometimes  the  connection  is  made  in  the  form  of  a  reversed 
curve  with  radii  about  the  same  as  the  radius  of  the  turnout 
curve  under  the  false  idea  that  this  saves  distance  between  the 
P.P.  and  the  T.C.  The  actual  difference  in  z3  for  such  a  connec- 
tion and  for  one  of  the  form  first  given  for  a  No.  10  turnout  is 
only  about  1.1  ft.,  an  inappreciable  amount  considering  the  rel- 
ative riding  qualities  of  the  two  layouts  for  heavy  road  engines. 


127.  Case  2.     Curved  Tracks. 

A  turnout  and  a  connection  to  a  parallel  track  on  a  curve 
may  be  staked  out  by  the  co-ordinates  given  in  Table  3.  Such 
a  layout  will  have  the  following  characteristics. 

1.  The  degree  of  curve  of  the  turnout  will  be  increased  or 
diminished  by  the  degree  of  the  main  curve,  depending  on 


FIG.  53. 


whether  the  siding  is  inside  or  outside  of  the  main  track  curve. 

2.  The  degree  of  the  connecting  curve  will  be  diminished  or 

increased  by  the  degree  of  the  main  curve  (approx.),  depending 


TURNOUTS,    CONNECTIONS,    AND    CROSSINGS         107 

on  whether  the  siding  is  inside  or  outside  of  the  main  track 
curve. 

3.  The  track  between  the  frog  point  and  the  beginning  of  the 
connecting  curve  will  become  a  curve  of  the  same  degree  (ap- 
prox.)  as  the  main  curve. 

On  curves  of  about  1°  this  method  is  both  satisfactory  and 
convenient,  but  on  sharper  curves,  or  in  case  it  is  desired  to 
maintain  a  frog  tangent,  it  will  be  necessary  to  compute  the 
connection  for  the  particular  case.  Two  general  cases  arise: 

(a)  Sidings  on  the  outside  of  the  main  curve. 
In  Fig.  53, 

OA     =  (R  +  ig)  cos  F 
AB     =  (R  +  ig)  sin  F 
AC     =  AB  +  P  +  K  =  OiH 
OOi    =  R  +  p  -  Ri 
OH    =  OA  -  (Rx  -  ig) 
In  triangle  2,         (OOO2   =  (OiH)2  +  (OH)2 

(R  +  p  -  RO2   =  (AC)2  +  (OA  +  ig  -  RO2 
expanding  and  reducing, 

(AC)2 

R'  =  2(OA  +  te  1  R  _  p)  +  (OA  +  *8  +  R  +  P)  •     (53) 
then 

AC 
sm  a  =  QQJ. 

Since  the  central  angle  a  and  the  radius  R\  are  now  known, 
the  length  of  the  connecting  curve  can  be  computed. 

The  curve  may  be  staked  out' either  by  deflection  angles  from 
either  end  (orient  how?)  or  by  offsets  from  the  main  track.  The 
latter  method  is  the  more  rapid  and  convenient. 

Since  the  connecting  curve  and  the  siding  curve  have  a  common 
tangent  and  since  the  angles  are  small,  the  offset  between  them 
at  any  point  is  equal  to  the  difference  of  their  tangent  offsets. 
Then  the  offset  from  the  center  line  of  the  main  track  to  the 
center  line  of  the  connecting  curve  at  any  point  is 
7 

p  — 5-  n2  (Di  —  Ds),  in  which  DI  and  Z)s  are  the  degrees  of 

o 

curve  of  the  connecting  track  and  the  siding,  respectively,  and 
n  is  the  distance  in  stations  along  the  connecting  curve  from  the 
point  where  it  joins  the  siding.  These  offsets  should  be  measured 


108 


TEXT   BOOK    ON   RAILROAD  SURVEYING 


radially  to  the  main  curve  and  the  distances  should  be  measured 
along  the  connecting  curve. 

(6)  Sidings  on  the  inside  of  the  main  curve. 


FIG.  54a. 

In  Fig.  54a, 

OA     =  (R  -  Jg)  cos  F 
AB     =  (R  -  Jg)  sin  F 
AC     =  AB  -  K  -  P  =  OH 
OOi    =  Ri  +  p  -  R 
OiH   =  Ri  +  ^g  -  OA 

In  triangle  2,        (OOi)2  =  (OH)2  +  (OiH)2 

(Ri  +  p  -  R)2  =  (AC)2  +  (Ri  +  Jg  -  OA)2 

expanding  and  reducing, 


then 


AC 

OOI 


If  Eq.  54  gives  a  positive  value  of  RL,  the  layout  is  as  shown 
in  Fig.  54a. 

If  a  negative  value  of  #1  is  obtained,  the  layout  is  as  shown  in 
Fig.  54b  and  the  value  of  RI  is  numerically  correct. 

If  (OA  —  %g)  -  R  —  p,  Eq.  54  gives  RI  =  infinity,  and  the 


TURNOUTS,    CONNECTIONS,    AND    CROSSINGS        109 

frog  tangent  produced  is  tangent  to  the  siding  as  shown  in 
Fig.  54c. 

These  connections  can  be  staked  out  by  offsets  as  explained 
under  case  (a).     It  is  to  be  noted,  however,  that  if  the  layout  is 


FIG.  54b. 


FIG.  54c. 


as  shown  in  Fig.  54b,  the  sum  of  the  tangent  offsets  must  be  used. 

PARALLEL  TRACKS.    CROSSOVERS. 

Since  a  crossover  is  equivalent  to  an  entry  to  a  siding,  and 
therefore  both  turnouts  will  be  run  over  at  practically  the  same 
speed,  there  is  no  valid  reason  for  making  one  turnout  flatter 
than  the  other,  even  when  the  crossover  is  between  a  main  track 


FIG.  55. 

and  a  siding.     A  crossover  should  therefore  always  have  frogs 
of  the  same  number  at  both  ends. 

128.  Case  1.     Straight  Tracks. 

(a)  Straight  track  between  frogs  of  the  same  number. 


110 


TEXT   BOOK   ON    RAILROAD    SURVEYING 


From  Fig.  55,  AC  =  P  cot  F 

then  x     =  AC  -  2gN  -  2P    .     .     .     .     (55) 

To  locate  the  crossover  one  P.F.  is  set  so  that  the  heel  or  toe 
will  be  at  a  regular  rail  joint.  The  other  P.F.  is  fixed  by  the 
distance  x.  These  are  the  only  stakes  ordinarily  required. 

Table  4  gives  the  distance  x  between  the  frog  points  for  track 
centers  from  12  to  16  ft.  for  turnouts  from  No.  7  to  15. 

TABLE  4 

Distances  along  main  track  between  frog  points  for  crossovers  between 
parallel  tracks.     Frogs  of  same  number,  straight  track  between  frogs. 


Track  Centers 

N 

F 

12.0 

12.5 

13.0 

13.5 

14.0 

14.5 

15.0 

15.5 

16.0 

7 

8°  10' 

17.14 

20.63 

24.11 

27.59 

31.08 

34.66 

38.05 

41.53 

45.02 

8 

7°  09' 

19.66 

23.64 

27.61 

31.62 

35.60 

39.59 

43  .  57 

47.56 

51  .  55 

9 

6°  22' 

22.05 

26.53 

31.05 

35.49 

39.97 

44.45 

48.93 

53.42 

57.90 

10 

5°  44' 

24.52 

29.50 

34.48 

39.46 

44.45 

49.43 

54.41 

59.39 

64.37 

12 

4°  46' 

29.90 

35.90 

41.90 

47.90 

53.89 

59.89 

65.89 

71.88 

77.88 

15 

3°  49' 

37.38 

44.88 

52  .  37 

59.86 

67.36 

74.85 

82.35 

89.84 

97.34 

(6)  Simple  curve  between  frogs  of  different  number. 
Sometimes  unequal  frogs  are  used  on  crossovers,  and  in  this 


FIG.  56. 


TURNOUTS,    CONNECTIONS   AND    CROSSINGS         111 

case  the  best  form  of  layout  to  use  is  a  connecting  curve  of  about 
the  same  degree  as  the  sharper  turnout,  tangent  at  the  heel  of  the 
flatter  frog.  This  will  place  a  short  piece  of  tangent  between 
the  curve  and  the  heel  of  the  sharper  frog  which  is  desirable. 

In  Fig.  56,  F,  Fi}  p,  g,  and  Ki  are  known.  R  is  chosen  to 
agree  with  the  turnout  curve  of  F. 

In  triangle  1,  the  hypotenuse  and  angle  FI  are  known.  Solve 
for  the  sides  d\  and  m. 

In  triangle  2,  the  altitude  n  =  p  —  m  and  the  angle  F  are 
known.  Solve  for  the  sides  d  and  j.  Then 

K  =  j  -  gN  -  R  tan  \  (F  -  FI) 
The  distance  along  the  main  tracks  between  frog  points  is 

x  =  (d  +  dO  -  (gN  +  P  +  gNi  +  Pi)      .     .     (56) 

129.  Case  2.     Curved  Tracks. 

It  can  be  shown  that,  if  a  crossover  is  located  between  curved 
tracks  with  the  same  frog  numbers  and  a  simple  curve  tangent 
to  the  heels  of  the  frogs,  the  distance  along  the  main  tracks  be- 
tween frog  points  is  practically  the  same  as  for  a  straight  cross- 
over between  straight  tracks;  and  also  that  the  degree  of  curve 
of  the  connection  is  practically  the  same  as  that  of  the  main 
tracks. 

Therefore,  a  crossover  on  a  curve  is  located  with  the  same 
dimensions  as  one  on  tangent.  It  may  be  desirable,  however,  to 
locate  points  on  the  connecting  curve.  Usually  one  point  at 
the  middle  is  sufficient,  and  it  will  be  seen  that  this  point  is 
midway  between  the  frog  points  and  on  the  center  line  between 
tracks. 

Crossovers  should  be  avoided  on  curves. 

INTERSECTING  TRACKS. 

130.  Case  1.    Both  Tracks  Straight. 

(a)  Connection  in  form  of  simple  curve. 

Given  in  Fig.  57,  the  intersection  angle,  /;  the  frog  angles, 
F  and  F\\  and  the  radius  of  the  connecting  curve. 

Required:  The  distances,  A  and  B,  from  the  intersection  of  the 
center  lines  to  points  opposite  the  frog  points. 


112 


TEXT   BOOK   ON    RAILROAD    SURVEYING 


In  triangle  1,  the  angles  are  known  and  one  side,  n,  is  readily 
obtainable.     Solve  for  r  and  m. 


FIG.  57. 

In  triangle  2,  one  side, .;  +  r,  and  the  angles  are  now  known, 
whence  k  and  I  can  be  computed. 
Then 

A  =  k  -  gNi  -  Pi 

B=Z+m     -gN-P 


(57) 


(b)  Connection  in  form  of  reversed  curve. 

Given  in  Fig.  58,  the  intersection  angle,  7;  the  frog  angles, 
F  and  Fi't  the  radii  of  the  reversed  curve,  R  and  Ri'}  and  the 
location  of  one  frog  point,  which  is  the  distance,  T,  from  the 
P.7.  at  A. 

Required:  The  central  angles,  a  and  6,  of  the  reversed  curve, 
the  distance,  T\t  from  the  P.I.  to  the  unknown  frog  point,  and 
the  distance  x  between  frog  points  measured  along  the  straight 
track. 

In  triangle  1  (CDB)  CD  and  angle  F  are  given.  Solve  for  CB 
and  DB,  whence  AB  and  OB  can  be  found. 

In  triangle  2  (ABE)  AB  and  the  angles  are  known.  Solve  for 
BE  and  AE. 

In  triangle  3  (OEG)  OE  and  the  angles  are  known.  Solve  for 
OG  and  GE. 

In  triangle  4  (LHQ)  LH  and  the  angles  are  known.  Solve 
for  HQ  and  LQ  =  JM. 


TURNOUTS,    CONNECTIONS,    AND    CROSSINGS         113 


In  triangle  5  ( HJOi)  HOi  and  the  angle  FI  are  known.  Solve 
for  OiJ  and  HJ. 

In  triangle  6  (OiOS)  OiO  =  R  +  Ri  and  OiS  =  OJ  +  JM 
+  GO  are  known.  Solve  for  angles  c  and  d,  whence  angles  a 
and  6  are  readily  determined. 

T!  =  AE  +  EG  -  OS  -  LM  +  gNi  +  PI      .     (58) 
x   =  T  -  AL  cos  I  -  gNi  -  Pi         .     .     .     .     (59) 

If  Ti  instead  of  T  is  known  the  plan  of  solution  is  the  same. 

If  the  tracks  do  not  run  to  an  intersection  or  if  it  is  inadvisable 
to  run  out  the  tangents  and  determine  7,  locate  the  point  D, 
set-up  and  turn  off  90°  from  the  frog  tangent,  and  measure  DE. 
Then  set-up  at  E  and  measure  DEL.  With  these  data  the  prob- 
lem can  be  solved  in  the  same  manner  as  before. 

131.  Case  2.     One  Straight  and  One  Curved  Track. 

From  Fig.  59, 

In  triangle  1,  the  hypotenuse  and  the  angle  F  are  known. 
Solve  for  the  two  sides.  Whence, 

In  triangle  2,  the  two  sides  can  be  obtained  by  arithmetic. 
Solve  for  the  hypotenuse  and  the  angle  a. 


114 


TEXT   BOOK    ON    RAILROAD    SURVEYING 


FIG.  59. 

In  triangle  3,  the  hypotenuse  and  the  angle  7  are  known. 
Solve  for  the  two  sides. 

In  triangle  4,  the  altitude  =  Ri  +  \Q  and  the  angle  F\  are 
known.  Solve  for  the  hypotenuse  and  the  base. 

In  triangle  5,  the  hypotenuse  and  the  angle  FI  are  known. 
Solve  for  the  sides  I  and  m.  Whence, 

In  triangle  6,  the  altitude  is  found  by  arithmetic  and  the 
hypotenuse  is  known  from  triangle  2.  Solve  for  the  angle  b. 
Then 

x   =  a  -  b  +  (90°  -  I)  -  (90°  -  F) 


and 


= a -b - I +  F 


(curve)  AB  (in  feet) 


(60) 


100 
D 


The  P.P.  is  set  at  the  distance  AB  -  P  from  A. 
From  triangles  3,  5,  and  6,  the  distances  AE,  m,  and  OG  are 
known.     Then  AC  is  found  by  arithmetic. 


TURNOUTS,    CONNECTIONS,    AND    CROSSINGS         115 

The  P.F.i  is  set  at  a  distance  AC  —  PI  from  A.  The  central 
angle  of  the  connecting  curve  z  =  I  -\-  x  —  F\  —  F. 

Obviously  such  a  connection  can  be  placed  in  any  of  the  four 
quadrants,  but  the  one  here  shown  gives  the  best  alinement 
and  should  be  used  when  possible. 

In  case  a  connection  is  to  be  placed  in  any  other  quadrant,  the 
plan  of  solution  is  identical  with  that  above,  and  if  followed 
exactly  no  difficulty  should  arise. 

132.  Case  3.    Both  Tracks  Curved. 

From  Fig.  60, 

In  triangle  1  (AOOi],  the  two  sides,  R  and  Ri,  and  the  included 
angle  /  are  known.  Solve  for  the  side  00 i  and  the  angles  a  and  6. 


In  triangle  2  (BEO),  the  hypotenuse  and  angle  F  are  known. 
Solve  for  the  sides  BE  and  EO. 

Then  in  triangle  3  (OOaG),  the  two  sides  are  found  by  arith- 
metic. Solve  for  the  angle  c  and  the  side  002. 


116  TEXT   BOOK   ON   RAILROAD    SURVEYING 

In  triangle  4  (CO\H)  the  hypotenuse  and  .the  angle  FI  are 
known.  Solve  for  the  sides,  CH  and  HOi. 

Then  in  triangle  5  (Oi02/),  the  two  sides  are  found  by  arith- 
metic. Solve  for  the  angle  d  and  the  side  OiOz. 

In  triangle  6  (00i02),  the  three  sides  are  now  known.  Solve 
for  the  angles  e  and  /.  Then 

x   =  b  -  f  +  c  -  (90°  -  F) 

=  b-f  +  c  +  F-90°      .     .     .     .     (61) 
and 

y    =  a  -  e  +  d  +  F!  -  90°          .     .     .     (62) 

curve  AB  (in  ft.)  =  — ~— ,  and  the'P.F.  is  set  at  a  distance 
AB  -  P  from  A. 

curve  AC  (in  ft.)  =  — FT*  an(^  the  P-f-i  is  set  at  the  dis- 
tance AC  —  PI  from  A. 

Then  the  central  angle  of  the  connecting  curve  is 

z=I+x  +  y-F-Fi         .     .     .     (63) 

Obviously  such  a  connection  can  be  placed  in  any  of  the  four 
quadrants,  but  the  one  here  shown  gives  the  best  alinement  and 
should  be  used  when  possible. 

In  case  a  connection  is  placed  in  any  other  quadrant  the  plan 
of  solution  is  identical  with  that  above,  and  if  followed  exactly 
no  difficulty  should  arise. 

3.  Crossings 

133.  A  crossing  is  a  device  used  at  the  intersection  of  two 
tracks.  It  consists  of  four  frogs  and  the  necessary  connecting 
rails.  Any  one  of  the  frogs  is  a  crossing  frog.  The  crossing 
angle  is  the  angle  between  the  center  lines  of  the  tracks  at  their 
point  of  intersection. 

Crossings  are  designated  as  single  curve,  double  curve,  or 
straight  according  as  one,  both,  or  neither  of  the  tracks  are 
curved. 

Crossings  are  usually  made  of  rolled  rails  fitted  together. 
When  the  crossing  angle  is  greater  than  about  25°,  the  various 
pieces  are  cut  to  fit  against  each  other  and  are  united  by  filling 
blocks  and  heavy  straps  well  bolted.  This  is  frequently  termed 


TURNOUTS,    CONNECTIONS,   AND    CROSSINGS        117 

solid  construction.  For  angles  under  about  25°,  regular  frog 
point  construction  is  used,  and  such  crossings  are  termed  frog 
crossings.  Distinguish  between  frog  crossing  and  crossing  frog. 

On  street  railway  work  hard  steel  (manganese)  centers  for 
frogs  have  been  used  for  many  years,  and  since  1905  such  con- 
struction has  been  growing  in  use  on  steam  roads  both  for 
standard  frogs  and  for  crossings.  (This  type  is  usually  designat- 
ed as  "  manganese  frogs  or  crossings.") 

There  have  been  numerous  attempts  to  construct  crossings 
with  a  revolving  section  or  turntable  at  each  frog  so  that  the 
wheels  will  have  no  flange  way  to  jump;  but  only  one  of  these 
seems  to  give  much  promise  of  success. 

The  end  frogs  of  a  frog  crossing  are  similar  to  a  standard  rigid 
frog  in  that  there  is  a  single  point  on  which  the  wheels  run. 
The  middle  frogs,  however,  have  two  running  points  and  are 
therefore  frequently  termed  "  double-pointed  frogs." 


FIG.  61.    Double  Slip  Switch. 

When  "  slip  switches  "  are  used,  the  crossing  is  made  to  a 
standard  frog  number,  and  if  located  at  an  interlocking  plant  the 
middle  frogs  are  frequently  made  with  movable  points.  That 
is,  with  movable  points  joined  in  pairs  and  moving  together 
similar  to  a  split  switch  in  such  a  way  that  the  wheels  have  a 
solid  bearing  and  no  flangeway  to  jump. 

A  "  slip  switch  "  or  "  combination  crossing  "  is  a  combination 
of  a  small  angle  crossing  with  a  pair  of  connecting  tracks  placed 
entirely  within  the  limits  of  the  crossing.  They  are  used  in  large 
yards  and  terminals  (Fig.  61). 

134.  Crossing  Data.  Very  few  railroads  construct  their  own 
crossings,  but  have  them  built  by  manufacturers  who  make 
a  specialty  of  such  work.  The  field  engineer  is  rarely  called 
on  to  compute  the  dimensions  of  a  crossing,  and  to  do  so  is  a 
waste  of  time  if  the  crossing  is  ordered  from  a  manufacturer. 


118 


TEXT   BOOK   ON   RAILROAD    SURVEYING 


It  is  far  more  important  that  the  makers  have  all  the  data,  and 
the  field  engineer  is  frequently  required  to  furnish  the  data.  The 
information  required  is: 

1.  The  crossing  angle. 

2.  The  gage  of  each  track. 

3.  The  curvature  —  degree  of  curve,  radii,  or  the  equivalent. 

4.  The  direction  of  curvature. 

5.  The  length  along  each  gage  line  from  one  gage  line  in- 
tersection (theoretical  P.P.)  to  the  nearest  rail  joint. 

6.  Length  over  all  along  each  gage  line. 

7.  The  height,  weight,  and  style  of  rail  of  which  the  crossing 
is  to  be  made. 

8.  The  height,  weight,  and  style  of  rail  in  intersecting  track 
if  offset  splices  are  to  be  furnished. 

9.  The  spacing  and  size  of  holes  for  splice  bars. 

10.  The  type  of  crossing,  etc.,  unless  covered  by  general 
specifications. 

This  information  can  best  be  given  by  means  of  a  small  sketch. 
Field  dimensions  should  be  taken  to  the  nearest  1/8"  (0.01  ft.). 

Occasionally  the  field  engineer  is  called  on  to  compute  the  di- 
mensions of  a  crossing.  The  values  which  are  required  are  the 
frog  angles,  FI,  F2,  F3,  F^  the  length  of  sides  along  the  gage 
lines,  and  the  two  diagonals.  The  computations  should  be  made 
with  sufficient  accuracy  to  give  results  which  are  correct  to  the 
nearest  1/16",  which  is  the  working  limit  of  the  manufacturers. 

The  solutions  of  the  three  kinds  of  crossings  are  outlined  as 
follows : 

135.  Straight  Crossings. 


FIG.  62. 


(a)   Unequal  Gages.     In  Fig.  62,  git  g2,  and  I  are  given,  and 
the  sides  and  diagonals  are  required. 


TURNOUTS,    CONNECTIONS,    AND    CROSSINGS        119 

In  triangle  1,  the  altitude  gi  and  the  angle  7  are  known. 
Solve  for  si  and  61. 

Similarly  solve  for  $2  and  62  in  triangle  2. 

In  triangle  3,  the  altitude  gi  and  the  base  s2  +  61  are  known. 
Solve  for  the  hypotenuse,  which  is  the  long  diagonal  of  the 
crossing. 

In  triangle  4,  the  altitude  02  and  the  base  si  —  b2  are  known. 
Solve  for  the  hypotenuse,  which  is  the  short  diagonal  of  the 
crossing. 

(6)  Equal  Gages.  In  this  case,  triangles  1  and  2  are  identical 
and  all  four  sides  are  equal.  The  angle  between  the  hypotenuse 
and  the  base  of  triangle  3  becomes  J7,  and  of  triangle  4  becomes 
90  —  ^7  which  simplifies  the  solution. 

136.  Single-curve  Crossings. 

Given,  in  Fig.  63,  0i,  02,  and  7  to  find  the  frog  angles,  length 
of  sides,  and  both  diagonals. 


FIG.  63 


Project  the  radius  on  a  perpendicular  to  the  straight  track. 

Then  OB  =  R  cos  7,  whence  OE  and  OC  are  known. 

In  triangle  FiCO,  the  base  and  hypotenuse  are  known.  Solve 
for  FiC  and  the  angle  FI. 

Similarly  solve  for  F2  and  FZC,  Fs  and  FSE,  and  F4  and  F*E 
from  corresponding  triangles.  Then  the  straight  sides  F\F* 
and  FzFi  can  be  found  by  subtraction.  The  curved  sides  FiFt 
and  F2F3  can  be  determined  since  their  radii  are  known  and 
their  central  angles  can  be  found  by  subtraction. 


120  TEXT   BOOK   ON   RAILROAD    SURVEYING 

The  long  diagonal  is  the  hypotenuse  of  a  triangle  whose  altitude 
is  g\  and  whose  base  is  F$E  —  FiC,  and  the  short  diagonal  is  the 
hypotenuse  of  a  triangle  whose  altitude  is  g\  and  whose  base  is 
F2C  -  F4E. 

137.  Double-curve  Crossings. 

In  Fig.  64,  #1,  R2,  gl}  g2)  and  /  are  given,  and  it  is  required  to 
find  the  frog  angles,  length  of  sides,  and  both  diagonals. 

In  triangle  A0i02,  two  sides  and  the  included  angle  are  known. 
Solve  for  the  distance  between  centers,  0\02. 

Then  in  the  four  triangles  formed  on  Oi02  by  the  radii  from 
Fi,  F2,  FS}  and  F4  the  three  sides  of  each  are  known,  and  the 


o, 


FIG.  64. 

angles  can  be  solved  for.  From  the  angles  at  0\  and  02  determine 
the  central  angles  of  the  arcs,  FiF2,  F2F3,  FSF4,  and  F4Flt 
from  which  the  lengths  of  the  arcs  can  be  computed  since  their 
radii  are  known.  The  long  diagonal  is  found  from  the  triangles 

or  Fi02Fz,  and  the  short  diagonal   from   the   triangles 

or 


APPENDIX 
.     THEORY  OF  SPIRAL 
138.  Derivation  of   o  and  A. 

From  definition,  d  =  ks]  fortheS.C.,  D  =  kS  .     .     .     .     (18) 

The  radius  of  curvature,  r  =  -r^ 

do 

Substituting  for  r  its  value,  -j- ,  and   then  for  d  its   value 
from  Eq.  18,  and  solving, 

ad  -  kldl 

5730 

kl2  ks2 


Integrating,  d  = 

in  which  o  is  in  circular  measure.     Changing  d  to  degrees, 

d  =  iks2    ........     (19) 

For  the  S.C.,  A  =  JkS2 

139.  Derivation  of  the  Co-ordinates  x  and  >. 

The  differential  equations  for  the  co-ordinates  of  a  spiral  are: 

dx  =vds  cos  d 
dy  =  ds  sin  d 

Expanding  the  sin  and  cos  into  series, 

/  fc        64 

dx  -  ds    1  -     "  +  T     ~ 


/*        <*3        A5  \ 

dy  =  ds  (d   -  IT  +  IF  -  etc-| 

?         E.     12-  ' 


ks2 
Substituting  TTT-^  for  d 


114.6  J 

/  k2s4 

dx  =  ds  ^1  —  ; 

121 


122  TEXT   BOOK   ON   RAILROAD    SURVEYING 

.    /  ks2  k3s6  ^ 

dy  =  ds  \U4f>  ~  6(114:6?  ~ 

k2s5  • 
Integrating,  x   =  s  -  T^-TJ-^  +  etc. 


ks3  k3s7 

"'  3(114.6)       42(114.6)3  * 

in  which  x  and  y  are  in  stations.     Reducing  to  feet  and  dropping 
small  terms, 

x   =  100  s  -  0.000762  k2s5    .     .     .     (23) 
y   =  0.291  ks3  -  0.00000158  k3s7    .     (22) 

140.  Derivation  of  a,  A,  b,  and  B. 

ks3      .  kV 

x        3X114.6     42X(114.6)3  +  etc' 
tan  a  =  —  =  -  V^2o5 

+  etc. 
f  etc. 


10  X  (114.6)2 
ks2  k3s6 


Substituting  d  for 


3  X  H4.6        (114.6)3  X  105 
ks2 


114.6 

t  I 


A          6  Is- 

tan-^-  =  —  +  "gT"  ^3  +  etc.  (tangent  series). 

1  24 

Whence,  tan  a  =  tan  --  o  —  Oz 


Reducing  to  degrees, 

tan  a  =  tan  -j  d  -  0.000000015  o3 

Investigating  the  second  term,  it  is  found  that  o  must  be  about 
25°  before  there  is  an  error  of  one  minute  in  a  by  considering  it  as 
^  $;  and  since  0  rarely  reaches  25°  and  any  error  of  direction 
is  corrected  at  the  S.C.,  it  is  sufficient  to  write 

a  =  -^-  d  =  lOks2  (in  minutes) 

....     (20) 

A  =  -^-A   =  10kS2  (in  minutes) 
<>  J 


THEORY  OF  SPIRAL  123 

and, 

b  =  y  3  =  2a,  and  B  =  J-  A  =  2A      .     .     (21) 

141.     Derivation  of  O. 

From  Fig.  65, 

o  =  Y  -  R  vers  A 

Substituting  for   Y  its  value,  0.291-  kS3,  for  R  its  value,  -j-g-, 
and  expanding  the  vers  series, 

o  =  0.291  kS3  -  0.00000158  k3S7  - 

5730  /       k2S4  k4S8      \ 


730  /_ 

kS    \2 


kS    \2  X  (114.6)2        24(114.6)4/ 

o  =  0.291  kS3  -  0.00000158  k3S7  -  5730 

/     kS3  k3S7      \ 

\2(114.6)2       24(114.6)V 

Reducing, 

o  =  0.0727  kS3  -  0.0000002  k3  S7 

Evidently  the  second  term  is  very  small,  hence, 

o  =  0.0727  kS3 (24) 

From  figure, 

t  =  X  -  R  sin  A 

Substituting  for  X  its  value  from  Eq.  23,  and  for  R  its  value, 

5730        ,  ,.       .,       . 

~r^-j  and  expanding  the  sin  series, 


t  =  100  s-  0.000762  k2S5 


-  ^~  /A  -  ^  +  etc.  ) 


-  1008  -  0.000762  k«S-  - 
Reducing, 


t  =  50  S  -  0.000127  k2S5   .....     (25) 


124 


TEXT   BOOK   ON    RAILROAD    SURVEYING 


Substituting  J  S  in  Eq.  23,  the  abscissa  of  the  middle  of  the 
spiral  is  50S  -  0.000024  k2S5.  The  difference  between  this  and 
Eq.  25  is  only  0.0001  k2S5,  showing  that  the  spiral  is  practically 
bisected  by  the  ordinate  to  the  T.C. 


FIG.  65. 

Substituting  J$  in  Eq.  24,  the  ordinate  for  the  middle  of  the 
spiral  is  0.0364  kS3. 

Jo  =  0.0363  kS3 

showing  that  o  is  bisected  by  the  spiral.     If  the  second  term  of 
Eq.  24  had  been  used  the  difference  would  have  been  still  less. 

142.  Osculating  Circle. 

The  fact  that  o  and  the  spiral  mutually  bisect  shows  that  the 
deflection  of  the  circle  from  the  spiral  between  the  S.C.  and  the 
T.C.  is  the  same  as  the  deflection  of  the  spiral  from  the  tangent 
in  half  its  length.  Since  the  S.C.  can  be  at  any  point  on  a 
spiral  of  given  k,  and  the  circular  curve  has  the  same  radius  as 
the  spiral  at  that  point,  it  follows  that  in  general  the  spiral 
departs  from  an  osculating  circle  at  the  same  rate  as  from  the 
initial  tangent. 


143.  Superelevation. 

From  mechanics, 


F  - 


Wv2 
2gR 


THEORY    OF    SPIRAL 


125 


where  F  is  the  centrifugal  force,  W  is  the  weight,  v  is  the  velocity 
in  feet  per  second,  R  is  the  radius  of  curvature,  and  g  is  the  ac- 
celeration of  gravity  (equal  to  16.08). 

In  Fig.  66,  AB  represents  the  horizontal  distance  between 
rail  heads,  which  for  standard  gage  is  about  4.9  ft.;  CB,  the 


FIG.  66. 


superelevation;   EH,  the  weight  W',  and  GE,  the  centrifugal 
force.     Then  from  similar  triangles, 


Substituting  the  above  value  of  F  in  this  equation,  we  have 


Wv2  _  We 
2gR  ~  AB 


whence, 


e  (in  feet)  = 


ABv2  _     4.9  v2 
2gR    ~  32.16  R 


5730, 


Reducing   v   to   miles   per   hour,  substituting  — g-  for  R,   and 
reducing, 

e  (in  inches)  =  0.00069  D  V2     ....     (28) 


TABLES 


128 


TABLE  5. — RADII. 


Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

0°    0' 

Infinite 

1°    0' 

5729.65 

2°   0/ 

2864.93 

3°    0' 

1910.08 

4°   0' 

1432.69 

1 

313:75. 

3 

5635.72 

1 

2841.26 

1       i 

1899.53 

1 

1426.74 

2 

171887. 

2 

5544.83 

2 

2817.97 

2 

1889.09 

2 

142(K85 

3 

114595. 

3 

5456.82 

3 

2795.06 

3 

1878.77 

3 

1415^01 

4 

85943.7 

4 

5371.56 

4 

2772.53 

4 

1868.56 

4 

1409.21 

5 

68754.9 

5 

5388.92 

5 

2750.35 

5 

1858.47 

5 

1403^46 

6 

57595.8 

6 

5208.79 

6 

2728  52 

6 

1848.48 

6 

1397176 

7 

49110.7 

7 

5131.05 

7 

2707.04 

7 

1838.59 

7 

1392.10 

8 

42971.8 

8 

5055.59 

8 

2685.89 

8 

1828.82 

8 

1386.49 

9 

38197.2 

9 

4982.33 

9 

2665.08 

9 

1819.14 

9 

1380  92 

10 

34377.5 

10 

4911.15 

10 

2644.58 

10 

1809.57 

10 

1375.40 

11 

31252.3 

11 

4841.98 

11 

2624.39 

11 

1800.10 

11 

1369.92 

12 

28G47.8 

12 

4774.74 

12 

2604.51 

12 

1790.73 

12 

1364.49 

13 

26444.2 

13 

4709.33 

13 

2584.93 

13 

1781.45 

13 

1359.10 

14 

24555.4 

14 

4645.69 

14 

2565.65 

14 

1772.27 

14 

1:353.75 

15 

22918.3 

15 

4583.75 

15 

2546.64 

15 

1763.18 

15 

1348.45 

16 

21485.9 

16 

4523.44 

16 

2527.92 

16 

1754.19 

16 

1343.15 

17 

20222.1 

17 

4464.70 

17 

2509.47 

17 

1745.26 

17 

1337.65 

18 

19098.6 

18 

4407.46 

18 

2491.29 

18 

1736.48 

18 

1332.77 

19 

18093.4 

19 

4351.67 

19 

2473.37 

19 

1727.75 

19 

1327  63 

20 

17188.8 

20 

4297.28 

20 

2455.70 

20 

1719.12 

20 

1322!  53 

21 

16370.2 

21 

4244.23 

21 

2438.29 

21 

1710.56 

21 

1317.46 

2:2 

15626.1 

22 

4192.47 

22 

2421.12 

22 

1702.10 

22 

1312.43 

23 

14946.7 

23 

4141.96 

23 

2404.19 

23 

1693.72 

23 

1307.45 

24 

14323.6 

24 

4092.66 

24 

2387.50 

24 

1685.42 

24 

1302  50 

25 

13751.0 

25 

4044.51 

25 

2371.04 

25 

1677.20 

25 

1297.58 

26 

13222.1 

26 

3997.49 

26 

2354.80 

26 

1669.06 

26 

1292  71 

27 

12732.4' 

27 

3951.54 

27 

2338.78 

27 

1661.00 

27 

1287.87 

28 

12277.7 

28 

3906.54 

28 

2322.98 

28 

1653.01 

28 

1283.07 

29 

11854.3 

29 

3862.74 

29 

2307.39 

29 

1645.11 

29 

1278  30 

30 

11459.2 

30 

3819.83 

30 

2292.01 

30 

1637.28 

30 

1273.57 

31 

11089.6 

31 

3777.85 

31 

2276.84 

31 

1629.52 

31 

1268.87 

32 

10743  0 

32 

3736.79 

32 

2261.86 

32 

1621.84 

32 

1264.21 

33 

10417.5 

33 

3696.61 

33 

2247.08 

33 

1614.22 

33 

1259^58 

34 

10111.1 

34 

3657.29 

34 

2232.49 

34 

1606.68 

34 

1254.98 

35 

9822.18 

35 

3618.80 

35 

2218.09 

35 

1599.21 

35 

1250.42 

36 

9549.311 

36 

3581.10 

36 

2203.87 

36 

1591.81 

36 

1245.89 

37 

9291.29 

37 

3544.19 

37 

2189.84 

37 

1584.48 

37 

1241.40 

38 

9046.75 

38 

3508.02 

38 

2175.98 

38 

1577.21 

38 

1236  94 

39 

8814.78 

39 

3472.59 

39 

2162.30 

39 

1570.01 

39 

1232.51 

40 

8594.42 

40 

3437.87 

40 

2148.79 

40 

1562.88 

40 

1228.11 

41 

8384.80 

41 

3403.83 

41 

2135.44 

41 

1555.81 

41 

1223  74 

42 

8185.16 

42 

3370.46 

42 

2122.26 

42 

1548.80 

42 

1219.40 

43 

7994.81 

43 

3337.74 

43 

2109.24 

43 

1541.86 

43 

1215  30 

44 

7813.11 

44 

3305.65 

44 

2096.39 

44 

1534.98 

44 

1210.82 

45 

7639.49 

45 

3274.17 

45 

2083.68 

45 

1528.16 

45 

1206.57 

46 

7473.42 

46 

3243.29 

46 

2071.13 

46 

1521.40 

46 

1202.36 

47 

7314.41 

47 

3212.98 

47 

2058.73 

47 

1514.70 

47 

1198.17 

48 

7162.03 

48 

31&3.23 

48 

2046.48 

48 

1508.06 

48 

1194.01 

49 

7015.8? 

49 

3154.03 

49 

2034.37 

49 

1501.48 

49 

1189  88 

50 

6875.55 

50 

3125.36 

50 

2022.41 

50 

1494.95 

50 

1185.78 

61 

6740.74 

51 

3097.20 

51 

2010.59 

51 

1488.48 

51 

1181.71 

62 

6611.12 

52 

3069.55 

52 

1998.90 

52 

1482.07 

52 

1177.66 

53 

6486.38 

53 

3042.39 

53 

1987.35 

53 

1475.71 

53 

1173.65 

54 

6366.26 

54 

3015.71 

54 

1975.93 

54 

1469.41 

54 

1169  66 

55 

6250.51 

55 

2989.48 

55 

1964.64 

55 

1463.16 

55 

1165.70 

56 

6138.90 

56 

2963.71 

56 

1953.48 

56 

1456.96 

56 

1161.76 

57 

6031.20 

57 

2938.39 

57 

1942.44 

57 

1450.81 

57 

1157.85 

58 

5927.22 

58 

2913.49 

58 

1931.53 

58 

1444.72 

58 

1153.97 

59 

5826.76 

59 

2889.01 

59 

1920.75 

59 

1438.68 

59 

1150  11 

60 

5729.65 

60 

2864.93 

60 

1910.08 

60 

1432.69 

60 

1146.28 

TABLE  5.— RADII. 


129 


1  — 
Deg 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

6°0 

1146.28 

6°0 

955.37 

7°0 

818.  G4 

8°0' 

716.34 

9°0/ 

G36.78 

I 

1142.47 

1 

952.72 

1 

816.70 

1 

714.85 

1 

635.61 

2 

1138.69 

2 

950.09 

2 

814.76 

2 

713.37 

2 

634.44 

3 

1134.94 

3 

947.48 

3 

812.83 

3 

711.90 

8 

633.27 

4 

1131.21 

4 

944  .  83 

4 

810.92 

4 

710.43 

4 

632.10 

5 

1127.50 

5 

942.29 

5 

809.01 

5 

708.96 

5 

630.94 

6 

1123.82 

6 

939.72 

6 

807.11 

6 

707.51 

6 

629.79 

7 

1120.16 

7 

937.16 

7 

805.22 

7 

706.05 

7 

628.64 

8 

1116.52 

8 

934.62 

8 

803.34 

8 

704.60 

8 

627.49 

9 

1112.91 

9 

932.09 

9 

801.47 

9 

703.16 

9 

626.  35 

10 

1109.33 

10 

929.57 

10 

799.61 

10 

701.73 

10 

625.21 

11 

1105.76 

11 

927.07 

11 

797.75 

11 

700.30 

11 

624.08 

12 

1102.22 

12 

924.58 

12 

795.91 

12 

G98.88 

12 

622.95 

13 

1098.70 

13 

922.10 

13 

794.07 

13 

697.46 

13 

621.82 

14 

1095.20 

14 

919.64 

14 

79^.24 

14 

696.05 

14 

620.70 

15 

1091.73 

15 

917.19 

15 

790.42 

15 

694.65 

15 

619.58 

16 

1088.28 

16 

914.75 

16 

788.61 

16 

693.24 

16 

618.47 

17 

1084.85 

17 

912.33 

17 

786.80 

17 

691.85 

17 

617.30 

18 

1081.44 

18 

909.92 

18 

785.01 

18 

690.46 

18 

616.25 

19 

107§.05 

19 

907.52 

19 

783.22 

19 

689.08 

19 

615  15 

20 

1074.68 

20 

905.13 

20 

781.44 

20 

687.70 

20 

614.05 

21 

1071.34 

21 

902.76 

21 

779.67 

21 

686.33 

21 

612.96 

22 

1068.01 

22 

900  .40 

22 

777.91 

22 

684.96 

22 

611.87 

23 

1064.71 

20 

898.05 

23 

776.15 

23 

G83.60 

23 

610.78 

24 

1061.43 

24 

895.71 

24 

774.40 

24 

682.25 

24- 

609.70 

25 

1058.16 

25 

893.39 

2:> 

772.66 

25 

680.89 

25 

608.62 

26 

1054.92 

26 

891.08 

26 

770.93 

26 

679.55 

26 

607.55 

27 

1051.70 

27 

'888.78 

27 

769.21 

27 

678.21 

27 

606.48 

28 

1048.48 

28 

886.49 

28 

767.49 

28 

676.88 

28 

605.41 

29 

1045.31 

29 

884.21 

29 

765.78 

29 

675.54 

29 

604.35 

30 

1042.14 

30 

881.95 

30 

764.08 

30 

674.22 

30 

603.29 

31 

1039.00 

31 

879.69 

31 

762.39 

31 

672.90 

31 

602.23 

32 

1035.87 

32 

877.45 

32 

700.70 

32 

671.59 

32 

601.18 

33 

1032.76 

33 

875.22 

33 

759.02 

33 

670.28 

33 

600.13 

34 

1029.67 

34 

873.00 

34 

757.35 

34 

668.98 

34 

599.09 

35 

1026.60 

85 

870.80 

35 

755.69 

35 

607.68 

35 

598.04 

36 

1023.55 

36 

868.60 

36 

754.03 

36 

666.39 

36 

597.01 

37 

1020.51 

37 

866.41 

37 

752.38 

37 

665.10 

37 

595.97 

38 

1017.49 

38 

864.24 

38 

750.74 

38 

603.82 

38 

594.94 

39 

1014.50 

39 

862.08 

39 

749.10 

39 

662.54 

39 

593.91 

40 

1011.51 

40 

859.92 

40 

747.48 

40 

661.26 

40 

592.89 

41 

1008.55 

41 

857.78 

41 

745.86 

41 

659.99 

41 

591.87 

42 

1005.60 

42 

855.65 

42 

744.24 

42 

658.73 

42 

590.85 

43 

1002.67 

43 

853.53 

43 

742.63 

43 

657.47 

43 

589.84 

44 

999.76 

44 

851.42 

44 

741.03 

44 

656.22 

44 

588.83 

45 

996.87 

45 

849.32 

45 

739.44 

45 

654.97 

45 

587.83 

46 

993.99 

46 

847.23 

46 

737.80 

46 

653.72    1 

46 

586.82 

47 

991.13 

47 

845.15 

47 

736.28 

47 

652.48 

47 

585.83 

48 

988.28 

48 

843.08 

48 

734.70 

48 

651.25 

48 

584.83 

49 

985.45 

49 

841.02 

49 

733.14 

49 

650.02 

49 

583.84 

50 

982.64 

50 

838.97 

50 

731.53 

50 

648.79 

50 

582.85 

51 

979.84 

51 

836.93 

51 

730.03 

51 

647.57 

51 

581.86 

52 

977.06 

52 

834.90 

52 

728.48 

52 

646.35 

52 

580.88 

53 

974.29 

53 

832.89 

53 

726.94 

53 

645.14 

53 

579.90 

54 

971.54 

54 

830.88 

54 

725.41 

54 

643.94 

54 

578.92 

55 

968.81 

55 

828.88 

55 

723.88 

55 

642.73 

55 

577.95 

56 

966.09 

56 

826.89 

56 

722.36 

56 

641.53 

56 

576.98 

57 

963.39 

57 

824.91 

57 

720.85 

57 

640.34 

57 

576.02 

58 

960.70 

58 

822.93 

58 

719.34 

58 

639.15 

58 

575.06 

59 

958.03 

59 

820.97 

59 

717.84 

59 

637.96 

59 

574.10 

60 

955.37 

60 

819.02 

60 

716.34 

60 

636.78 

60 

573.14 

130 


TABLE  5.— RADII. 


Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

10°  0' 

573.14 

12»0' 

477.68 

14°  0' 

409.32 

16°  0' 

358.17 

18°  0' 

318.39 

2 

571.24 

2 

476.36 

2 

408.35 

2 

357.43 

2 

317.80 

4 

569.35 

4 

475.05 

4 

407.38 

4 

356.69 

4 

317.22 

6 

567.47 

6 

473.74 

6 

406.42 

6 

355.95 

6 

316.63 

8 

565.60 

8 

472.44 

8 

405.46 

8 

355.21 

8 

316.05 

10 

563.75 

10 

471.15 

10 

404.51 

10 

354.48 

10 

315.47 

12 

561.91 

12 

469.86 

12 

403.56 

12 

353.75 

12 

314.89 

14 

560.08 

14 

468.58 

14 

402.61 

14 

353.03 

14 

314.32 

16 

558.26 

16 

467.31 

16 

401.67 

16 

352.30 

16 

313.75 

18 

556.45 

18 

466.04 

18 

400.74 

18 

351.58 

18 

313.18 

20 

554.66 

20 

464.78 

20 

399.80 

20 

350.86 

20 

312.61 

22 

552.88 

22 

463.53 

22 

398.88 

22 

350.15 

22 

312.04 

24 

551.11 

24 

462.29 

24 

397.95 

24 

349.44 

24 

311.47 

26 

549.35 

26 

461.05 

26 

397.03 

26 

348.72 

26 

310.91 

28 

547.60 

28 

459.82 

28 

396.13 

28 

348.02 

28 

310.35 

30 

545.87 

30 

458.59 

30 

395.21 

30 

347.32 

30 

309.79 

32 

544.14 

32 

457.88 

32 

394.30 

32 

346.62 

32 

309.23 

34 

542.42 

34 

456.16 

34 

393.40 

84 

345.93 

34 

308.68 

36 

540.72 

36 

454.96 

36 

392.50 

36 

345.23 

36 

808.13 

38 

539.03 

38 

453.76 

38 

391.61 

38 

344.54 

38 

307.58 

40 

537.34 

40 

452.57 

40 

390.72 

40 

343.85 

40 

307.03 

42 

535  67 

42 

451.38 

42 

389.83 

42 

343.16 

42 

306.48 

44 

534.01 

44 

450.20 

44 

388.95 

44 

342.48 

44 

305.93 

46 

532.36 

46 

449.02 

46 

388.07 

46 

341.80 

46 

305.39 

48 

530.71 

48 

447.86 

48 

387.20 

48 

841.12 

48 

304.85 

50 

529.08 

50 

446.69 

50 

386.33 

50 

340.45 

50 

304.31 

52 

527.46 

52 

445.54 

52 

385.47 

52 

339.78 

52 

303.77 

54 

525.85 

54 

444.39 

54 

384.60 

54 

339.11 

54 

303.24 

56 

524.25 

56 

443.24 

56 

383.75 

56 

338.44 

56 

302.70 

58 

522.65 

58 

442.11 

58 

382.89 

58 

337.77 

58 

302.17 

11°  0' 

521.07 

18°  0' 

440.97 

16°  0' 

382.04 

17°0' 

337.11 

19°  0' 

301.64 

2 

519.50 

2 

439.85 

2 

381.19 

2 

336.45 

2 

301.12 

4 

517.93 

4 

438.73 

4 

380.35 

4 

335.80 

4 

800.59 

6 

516.38 

6 

437.61 

6 

379.51 

6 

335.14 

6 

800.07 

8 

514.84 

8 

436.50 

8 

378.68 

8 

334.49 

8 

299.  E4 

10 

513.30 

10 

435.40 

10 

377.84 

10 

333.84 

10 

299.02 

12 

511.77 

12 

434.30 

12 

377.0:2 

12 

333.19 

12 

298.50 

14 

510.26 

14 

433.21 

14 

376.19 

14 

332.55 

14 

297.99 

16 

508.75 

16 

432.12 

16 

375.37 

16 

331.91 

16 

297.47 

18 

507.25 

18 

431.04 

18 

374.55 

18 

331.  27 

18 

296.96 

20 

505.76 

20 

429.96 

20 

373.74 

20 

330.63 

20 

296.45 

22 

504.28 

22 

428.98 

22 

372.93 

22 

330.00 

22 

295.94 

24 

502.80 

24 

427.82 

24 

372.12 

24 

329.37 

24 

295.43 

26 

501.34 

26 

426.76 

26 

371.32 

26 

328.74 

26 

294.92 

28 

499.88 

28 

425.71 

28 

370.52 

28 

328.11 

28 

294  42 

30 

498.43 

30 

424.66 

30 

369.72 

30 

327.48 

30 

293.91 

32 

496.99 

32 

423.61 

32 

368.93 

32 

326.86 

32 

293.41 

34 

495.56 

34 

422.57 

34 

368.14 

34 

326.24 

34 

292.91 

36 

494.14 

36 

421.54 

36 

367.35 

3G 

325.62 

36 

292.41 

38 

492.73 

38 

420.51 

38 

306.57 

38 

325.01 

38 

291.92 

40 

491.32 

40 

419.49 

40 

365.79 

40 

324.40 

40 

291.42 

42 

489.92 

42 

418.47 

42 

365.01 

42 

323.79 

42 

290.93 

44 

488.53 

44 

417.45 

44 

364.24 

44 

323.18 

44 

290.44 

46 

487.15 

46 

416.44 

46 

863.47 

46 

322.57 

40 

289.95 

48 

485.77 

48 

415.44 

48 

362.70 

48 

321.97 

48 

289.46 

50 

484.40 

50 

414.44 

50 

361.94 

50 

321.37 

50 

288.98 

52 

483.05 

52 

413.44 

52 

361.18 

52 

320.77 

52 

288.49 

54 

481.69 

54 

412.45 

54 

360.42 

54 

320.17 

54 

288.01 

56 

430.35 

56 

411.47 

56 

359.67 

56 

319.57 

56 

287.53 

58 

479.01 

58 

410.49 

58 

358.9-3 

58 

318.98 

58 

287.05 

60 

477.68 

60 

409.51 

60 

358.17 

60 

318.39 

60 

28(5.57 

131 


TABLE  6.  —FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 

The  Long  Chords,  Mid-Ordinates,  Externals,  and  Tangent 
Distances  of  this  table  are  for  a  curve  of  5730  feet  radius.  To 
find  the  corresponding  functions  of  any  other  curve  divide  the 
tabular  values  by  the  degree  of  curve. 

For  metric  curves  having  20-metre  chords,  multiply  tne  degree 
by  5  and  enter  the  table  with  the  result  as  a  value  of  Z>,  the  tabu- 
lar values  being  taken  as  metres  instead  of  feet 

Thus  for  a  1°  30'  metric  curve  having  I—  45°  the  tangent  dis- 
tance is  T  =  — - — '—  =  316.45  metres.  Again,  suppose  J=  38° 
1 .  o  X  5 

and   the   long   chord  =  373.1  m.  known  and  D  required.     The 

tabular  L.  C.  is  3731  m.;  therefore  D  =      3731-°     =  2°  0> 

3.3.1  X  5 


0° 

1° 

L.  C.       M.           E.           T. 

L.  C.        M.           E.           T. 

0 

0.00     0.000      0.000        0.00 

100.00     0.218     0.218       50.00 

0 

2 

3.33     0000      0.000        1.67 

103.33     0.233     0.233       51.67 

2 

4 

6.6?     0  001      0.001        3.33 

106.66     0.248     0.248       53.33 

4 

6 

10.00     0.002      0.002        5.00 

110.00     0.264     0.264       55.00 

6 

8 

13.33     0.004      0.004        6.67 

113.33     0.280     0.280       56.67 

8 

10 

16.67     0.006      0.006        8.33 

116.66     0.297     0.297       58.33 

10 

12 

20.00     0.009      0.009      10.00 

120.00     0.314     0.314       60.00 

12 

14 

23.33     0.012      0.012      11.67 

123.33     0.332     0.332       61.67 

14 

16 

26.67     0.015      0.015      13.33 

126.66     0.350     0.350       63.33 

16 

18 

30.00     0.019      0.019      15.00 

130.00     0.368     0.368       65.00 

18 

20 

33.33     0.024      0.024      16.67 

133.33     0.388     0.388       66.67 

20 

22 

36.67     0.029      0.029      18.33 

136.66     0.407     0.407       68.33 

22 

24 

40.00     0.035      0.035-    20.00 

140.00     0.427     0.427       l~0.00 

24 

26 

43.33     0.041      0.041      21.67 

143.33     0.448     0.448       "1.67 

26 

28 

46.67     0.048      0.048      23.33 

146.66     0.469     0.469       ^3.33 

28 

30 

50.00     0.054      0.054      25.00 

15000     0.491      0.491        "5.00 

30 

32 

53  33     0.062      0.062      26.67 

153.33     0.513     0.513       ^6.67 

32 

34 

56  67     0.070      0.070      28.33 

156.66     0.536     0.536       78.33 

34 

36 

6000     0.079      0.079      30.00 

160.00     0.559     0.559       80.00 

36 

38 

63.33     0.088      0.088      31.67 

163.33     0.582     0.582       81.67 

38 

40 

66.67     0.097      0.097      33  33 

16666     0.606     0.606       83.33 

40 

42 

70.00     0.107      0.107      35.00 

170.00     0.630     0.630       85.00 

42 

44 

73.33     0.117      0.117      36.67 

173.33     0.655     0.655       86.67 

44 

46 

76.6?     0.128      0.128      38.33 

176.66     0.681      0.681       88.33 

46 

48 

80.00     0.140      0.140      40.00 

180.00     0.706     0.706       90.00 

48 

50 

83.33     0.151       0.151      41.67 

183.33     0733     0.733       91.67 

50 

52 

86  67     0.164      0.164      43.33 

186.66     0.760     0.760       93.33 

52 

54 

9000     0  176      O.J76      45.00 

190.00     0.788     0.788       95.00 

54 

56 

93  33     0.190      0.190      46.6? 

193.33     0.815     0.815       96.67 

56 

58 

96.67     0.204      0.204      4833 

196.66     0.844     0.844       98.33 

58 

tJB 

10000     0.218      0.218      5000 

199.98     0.873     0  873     100.00 

60 

132    6.— FUNCTIONS   OF    A   ONE-DEGREE   CURVE. 


2° 

3° 

4 

L.  C.         M.          E.          T. 

L.  C.         M.          E.           T. 

r 

0 

"199.98      0.873      0.873    100.00 

299.96      1.964      1.964    150.07 

0 

2 

203.31      0.902      0.902    101.67 

3U3.29      2.008      2.009    151.74 

2 

4 

206.64      0  932      0.932    103.34 

306.62      2.053      2.054    153.41 

4 

6 

209.97      0.962      0.962    105.01 

309.95      2.098      2.099    155.08 

6 

8 

213.31      0.993      0.993    106.68 

313.29      2.143      2.144    156.75 

8 

10 

216.64      1.024      1.024    108.35 

316.62      2.188      2.189     158.42 

10 

12 

219.97      1.056      1.056    110.02 

319.95      2.235      2.236    160.09 

12 

14 

223.30      1.088      1.088    111.69 

323.28      2.282      2.283    161.76 

14 

16 

226.64      1.121      1.121     113.36 

326.62      2.329      2.330     163.43 

16 

18 

229.97      1.154      1.154    115.02 

329.95      2.376      2.377    165.09 

18 

20 

233.30      1.188      1.188    116.69 

333.28      T.424      2.425    166.76 

20 

22 

236.63      1.222      1.222    118.36 

336.61      2.473      2.474    168.43 

22 

24 

239.97      1.256      1.256    120.03 

339.95      2.523      2.523    170.10 

24 

26 

243.30      1.292      1.292    121.70 

343.28      2.572      2.573    171.77 

26 

28 

246.63      1.328      1.328    123.37 

346.61      2.622      2.623    173  44 

28 

30 

249.96      1.364      1.364    125.03 

349.94      2.672      2.673    175.10 

30 

32 

253.29      1.399      1.399    126.70 

353.27      2.724      2.725    176.72 

32 

34 

256.62      1.437      1.437    128.37 

356.60      2.776      2.777    178.39 

34 

36 

259.96      1.475      1.475    130.04 

359.94      2.828      2.829    18006 

36 

38 

263.29      1.513      1.513    131.71 

363.27      2.880      2.881     181.73 

38 

40 

266.62      1.552      1.552    133.38 

366  60      2  933      2.934    183.40 

40 

42 

269.96      1.592      1.592    135.05 

369.94      2.987      2.988    185.07 

42 

44 

273.29      1.632      1.632    136.72 

373.27      3.042      3  043    186.74 

44 

46 

276.62      1.672      1.672    138.38 

376.60      3.096      3097    188.40 

46 

43 

279.96      1.712      1.712    140.05 

379.94      3.151      3.152    190.07 

48 

50 

283.29      1.752      1.752    141.72 

383.27      3.206      3.207    191.74 

50 

52 

286.62      1.794      1.794    143.39 

38660      3.263      3.264    193.41 

52 

54 

289.96      1.836      1.836    145.06 

389.94      3.320      3.321     195.08 

54 

56 

293.29      1.878      1.878    146.73 

393.27      3.377      3378    196.75 

56 

58 

296.62      1.921      1.921     148.40 

396.60      3.434      3.435    198.42 

58 

60 

299.96      1.964      1.964    150.07 

399.94      3.491      3.492    2uO  09 

60 

4° 

6° 

L.  C.         M.           E.          T. 

L.  C.         M.          E.          T. 

0 

399.94      3.491      3.492    200.09 

499.88      5.454      5.459    250.17 

0 

2 

403.27      3.550      3.551    201.70 

503.21      5.527      5.o33    251.84 

2 

4 

406.60      3.609      3.610    203.43 

506.54      5.601      5.607    253.51 

4 

6 

409.93      3.668      3.670    205.10 

509.87      5.675      5.681    255.18 

6 

8 

413.26      3.727      3.730    206.77 

513.20      5.749      5.755    256.85 

8 

10 

416.59      3.787      3.790    208.44 

51653      5.823      5.829    258.52 

10 

12 

419.92      3.848      3.851    210.11 

519.86      5.899      5.905    260.20 

12 

14 

423.26      3.910      3.913    211.77 

523.19      5.975      5981    261.86 

14 

16 

426.59      3.972      3.975    213.45 

526.52      6.052      6.058    263.54 

16 

18 

429.92      4.034      4.037    215.11 

529.85      6.129      6.135    265.20 

18 

20 

433.25      4.096      4.099    216.78 

533.18      6.206      6.212    266.87 

20 

22 

436.58      4.160      4.163    218.45 

536.51      6.284      6.290    208  54 

22 

24 

439.91      4.224      4.227    220.12 

539.  81      6.362      6.369    270.21  ' 

24 

26 

443.24      4.288      4.291     221.79 

543.17      6.441      6.448    271.88 

26 

28 

446.58      4.353      4.356    223.46 

546.50      6.520      6.527    273.54 

28 

30 

449.91      4.418      4.421     225.13 

549.83      6.599      6.606    275.21 

30 

32 

453.24      4.484      4.487    226.80 

553  17      6.680      6.687    276.88 

32 

34 

456.57      4.550      4.554    228.47 

556.50      6.761      6.768    278.55 

34 

36 

459.90      4.617      4.621    230.14 

559.83      6.842      6.849    280.23 

36 

38 

463.23      4.684      4.688    231.81 

563.16      6.923      6.931    281.90 

38 

40 

466.56      4.751      4.755    233.48 

566.49      7.005      7.013    283.57 

40 

42 

469.89      4.820      4.824    235.15 

569.82      7.088      7.096    285.24 

42 

44 

473.23      4.889      0.893    236.82 

573.15      7  171      7.180    286.91 

44 

46 

47656      4.958      4962    238.48 

576.48      7.255      7.264    288.59 

46 

48 

479.89      5.027      5.031    240.15 

579.81      7.339      7.348    290.26 

48 

50 

483.22      5.096      5100    241.82 

583.14      7423      7.432    291.93 

50 

52 

486.55      5.167      5.171    243.49 

586.47      7.508      7.517    293.60 

52 

54 

489.88      5.238      5.243    245.16 

589.80      7.593      7.603    295.27 

54 

56 

493.21       5.310      5.315    246.83 

593.13      7.678      7  689    296  95 

56 

58 

49654      5.382      5.387    248.50 

596.46      7.764      7.775    298.62 

58 

60 

499.88      5  454      5.459    250.17 

599.80      7.850      7  861     300.30 

60 

6  —FUNCTIONS  OF  A  ONE-DEGREE  CURVE.     133 


i                              6° 

7° 

L  C.          M.            E          T. 

L.  C.         M.           E.           T. 

0 

599.80      7.850      7.861    300.30 

699.60     10.69     10.71     350.44 

0 

2 

603.13      7.940      7.951    301.97 

702.93     10.79     10.81     352.11 

2 

4 

606  46      8.030      8.041    303.64 

706.26     10.90     10.92     353.79 

4 

6 

609.78      8.120      8.131    305.  31 

709.58     11.00     11.02     355.46 

6 

8 

613.11      8.210      8.  2:il    306.98 

712.91     11.11     11.13     357.13 

8 

10 

616  44      8.300      8.311    308.65 

716.24     11.21     11.23     358.81 

10 

12 

619.76      8.390      8.401    310.32 

719.56     11.31     11.33     360.48 

12 

14 

623.09      8.480      8,491    311.99 

722.89     11.42     11.44     362.15 

14 

16 

626.42      8.570      8.581    313.66 

726.21     11.52     11.54     363  83 

16 

13 

629.74      8.660      8.671    315.33 

729.53     11.63     11.65     365.50 

18 

20 

633.07      8.750      8.761    317.00 

732.86     J1.73     11.75     367.17 

20 

'22 

63(5.40      8.844      8.856    318.67 

736.19     11.84     11.86     368.85 

22 

24 

639.72      8.939      8.951    320.34 

739.51     11.95     11.97     370.52 

24 

26 

643.05      9.033      9.046    322.01 

742.84     12.06     12.08     372.19 

26 

28 

646.38      9.128      9.141    323.68 

746.17     12.17     12.19     373.86 

28 

30 

649.70      9.222      9.236    325.35 

749.49     12.27     12.30     375.54 

30 

32 

653.03      9.317      9.331    327.02 

752.82     12.38     12.41     377.22 

32 

34 

656.36      9.411      9.426    328.69 

756.15     12.49     1252     378.89 

34 

36 

659.69      9.506      9.521    330.37 

759.47     12.60     12.63     380  57 

36 

38 

663.02      9.600      9.616    332.04 

762.80     12.71     12.74     382.24 

38 

40 

666.34      9.695      9.712    333.71 

766.13     12.82     12.85     383.92 

40 

42 

669.67      9.794      9.812    335.38 

769.45     12.93     12.96     385.60 

42 

44 

673.00      9.894      9.912    337.05 

772.78     13.04     13.08     387.27 

44 

46 

676.32      9.993    10.01      338.73 

776.11     13.15     13.19     388.95 

46 

48 

679.65    10.09      10.11      340.40 

779.43     13.26     13.31     390.62 

48 

50 

682.98    10.19      10.21      342.07 

782.76     13.37     13.42     392.30 

50 

52 

686.30    10.29      10.31      343.74 

786.09     13.48     13.53     393.98 

52 

54 

689.63    10.39      10.41      345.41 

789.41      13.59     13.65     395.65 

54 

56 

692.96    10.49      10.51      347.08 

792.74     13.70     13.76     397.33 

56 

58 

696.28    10.59      10.61      348.76 

796.07     13.81     13.88     399.01 

58 

60 

699  60    10.69      10.71      850.44 

799.40     13.96     13.99     400.70 

60 

8° 

9° 

L.  C.        M.           E.           T. 

L.  C.        M.           E.           T. 

/ 

0 

799.40     13.96      13.99     400.70 

899.10     17.66     17.71    450.95 

0 

2 

802.72     14.07      14.10     402.37 

902.42     17.79     17.84    452.63 

2 

4 

806.04     14.19      14.22     404.05 

905.74     17.92     17.98    454.31 

4 

6 

809.37     14.31       14.34     405.72 

909.07     18.06     18.11    455.98 

6 

8 

812.69     14  43      14.46     407.39 

912.39     18.19     18.25    457.66 

8 

10 

816.01     14.55      14.58     409.06 

915.71      18.32     18.38    459.34 

10 

12 

819.34     14.66      14.70     410.74 

919.04     18.46     18.52    461.02 

12 

14 

822.66     14.78       14.82     412.41 

922.36     18.59     18.65    462.70 

14 

16 

825.98     14.90       14.94     414.  OS 

925.68     18.72     18.79    464.37 

16 

18 

829.31     15.02      15.06     415.75 

929.01     18.86     18.92    466.05 

18 

20 

832.63     15.14      15.18     417.43 

932.33     18.99     19.06    467.73 

20 

22 

83595     15.26      15.30     419.10 

935.65     19.12     19.19    469.41 

22 

24 

839.28     15.38      15.43     400.77 

938.98     19.26     19.33    471.08 

24 

26 

842.60     15.51       15.55     422.45 

942.30     19.40     19.47    472.76 

26 

28 

845.92     15.63       15.68     424.12 

945.62     19.54     19.61    474.43 

28 

30 

849.25     15.75      15.80     425.79 

948.95     19.68     19.75    476.10 

30  • 

32 

852.57     15.88       15.93     427.47 

952.27     19.82     19.89    477.78 

32 

34 

855.89     16.00       16.05     429.15 

955.59     19.96     20.03    479.46 

34 

36 

859.22     16.12      16.18     430.82 

958.92     20.10     20.17    481.14 

36 

38 

862.54     16.25      16.30     432.50 

962.24     20.24     20.31    482.83 

38 

40 

865.86     16.38      16.43     434.18 

965.56     20.38     20.45    484.51 

40 

42 

869.19     16.50       16.55     435.86 

968.89     20.52     20.59    486.19 

42 

1    44 

872.51      16.63       16.68     437.54 

972.21     20.66     20.74    487.87 

44 

46 

875.83     16.76      16.81     439.21 

975.53     20.80     20.88    489.56 

46 

1    48 

879.16     16.89      16.94     440.89 

97886     20.94     21.03    491.24 

48 

50 

882.48     17.02      17.07     442.57 

982.18     21.09     21.17    492.92 

50 

52 

885.80     17.14      17.19     444.25 

985.50     21.23     21.31    494.60 

52 

54 

889.13     17.27      17.32     445.93 

988.83     21.37     21.46    496.28 

54 

56 

892.45     17.40       17.45     447.60 

992.15     21.51     21.60    497.96 

56 

58 

895.77     17.53       17.58     449.28 

995.47     21.65     21.75    499.65 

58 

60 

899.10     1766       17.71      450.95 

998.80     21.80     21.89    501.32 

60 

134       6.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


10° 

11° 

L  C.        M.          E.         T. 

L.  C.        M.           E.         T. 

0 

998.8      21.80      21.89    501.32 

1098.4     26.38      26.50     551.74 

0 

2 

1002.1      21.94      22.03    503.00 

1101.7     26.54      26.66     553.42 

2 

4 

1005.4      22.09      22.18    504.68 

1105.0     26.70      26.83     555.10 

4 

6 

1008.8      22.24      22.33    506.36 

1108.3     26.86      26.99     556.78 

6 

8 

1012.1      22.39      22.48    508.04 

1111.7     27.02      27.16     558.46 

8 

10 

1015.4      22.54      22.63    509.  ".2 

1115.0     27.19      27.32     560.14 

10 

12 

1018.7      22.68      22.78    511.40 

1118.3     27.35      27.48     561.82 

12 

14 

1022  0      22.83      22.93    513.08 

1121.6     27.51      27.65     563.50 

14 

16 

1025.4      22.98      23.08    514.76 

1124.9     27.67      27.81     565.18 

16 

18 

1028.7      23.13      23.23    516.44 

1128.2     27.83      27.98     566.86 

18 

20 

1032.0      23.28      23.38    518.12 

1131.6     28.00      28.14     568.54 

20 

22 

1035.3      23.43      23.53    519.80 

1134.9     28.17      28.30     570.22 

22 

24 

1038.6      23.58      23.68    521.48 

1138.2     28.34      28.47     571.90 

24 

26 

1042.0      23.73      23.84    523.16 

1141.5     28.50      28.64     573.58 

26 

28 

1045.3      23.88      23.99    524.85 

1144.8     28.67      28.81     575.27 

28 

80 

1048.6      24.04      24.14    526.53 

1148.1     28.84      28.98     576.95 

30 

32 

1051.9      24.19      24.30    528.21 

1151.5     29.00      29.14     578.63 

32 

34 

1055.2      24.34      24.45    529.89 

1154.8     29.17      29.31     580.32 

34 

36 

1058.6      24.49      24.60    531.57 

1158.1     29.34      29.48     582.00 

36 

38 

1061.9      24.64      24.76    533.25 

1161.4     29.50      29.65     583.69 

38 

40 

1065.2      24.80      24.91     534.93 

1164.7     29.67      29.82     585.87 

40 

42 

1068.5      24.95      25.06    536.61 

1168.0     29.84      29.99     587.05 

42 

44 

1071.8      25.11      25.22    538.29 

1171.4     30.01      30.17     588.74 

44 

46 

1075.2      25.27      25.38    539.97 

1174.7     30.18      30.34     590.42 

46 

48 

1078.5      25.43      25.54    541.65 

1178.0     30.35      30.52     592.11 

48 

50 

1081.8      25.59      25.70    543.33 

1181.3     30.53      30.69     593.79 

50 

52 

1085.1      25.74      25.86    545.01 

11S4.6     30.70      30.86     595.47 

52 

54 

1088.4      25.90      26.02    546.  G9 

1187.9     30.87      31.04     597.16 

54 

56 

1091.8      26.06      26.18    548.37 

1191.3     31.04      31.21     598.84 

56 

58 

1095.1      26.22      26.34    550.06 

1194.6     31.21      31.39     600.53 

58 

60 

1098.4      26.38      26.50    551.74 

1197.9     31.39      31.56     602.22 

60 

1*2° 

13° 

L.  C.        M.           E.         T. 

L.  C.        M.          E.         T. 

-0 

1197.9      31.39      31.56    602.22 

1297.3     36.83      37.07     652.87 

0 

2 

1201.2      31.57      31.73    603.91 

1300.6     37.02      37.26     654.56 

2 

4 

1204.5      31.74      31.91    605.60 

1303.9     37.21      37.46     656.25 

4 

6 

1207.8      31.93      32.09    607.28 

1307.2     37.40      37.65     657.93 

6 

8 

1211.1       32.09      32.27    608.97 

1310.5     37.59      37.85     659.62 

8 

10 

1214.5      32.27      32.45    610.66 

1313.8     37.79      38.04     661.31 

10 

12 

1217.8      32.45      32.63    612.35 

1317.2     37.98      38.23     663.00 

12 

14 

1221.1      32.62      32.81    614.04 

1320.5     38.17      38.43     664.69 

14 

16 

1224.4      32.80      32.99    615.72 

1323.8     38.36      38.62     666.37 

16 

18 

1227.7      32.97      33.17    617.41 

1327.1     38.55      38.82     668.06 

18 

20 

1231.0      33.15      33.35    619.10 

1330.4     38.75      39.01      669.75 

20 

22 

1234.3      33.33      33.53    6v!0.79 

1333.7     38.95      39.20     671.44 

22 

24 

1237.7      33.51      33.72    622.48 

1337.0     39.15      39.40     673.13 

24 

26 

1241.0      33.69      33.90    624.16 

1340.3     39.35      39.60     674.81 

26 

28 

1244.3      33.87      34.09    6^5.85 

1343.6     39.54      39.80     676.51 

28 

30 

1247.6      34.06      34.27    627.55 

1346.9     39.74      40.00     678.20 

30 

32 

1250.9      34.24      34.45    629.24 

1350.3     39.94      40.19     679.89 

32 

34 

1254.2      34.42      34.64    630.93 

1353.6     40.13      40.39     681.58 

34 

36 

1257.5      34.60      34.82    632.61 

1356.9     40.33      40.59     683.26 

36 

38 

1260.8      34.78      35.01    634.30 

1360.2     40.52      40.79     684.95 

38 

40 

1264.2      3497      35.19    635.99 

1363.5     40.71      40.99     686.64 

40 

42 

1267.5      35.16      35.37    637.68 

1366.8     40.91      41.19     68833 

42 

44 

1270.8      35.34      35.56    639.37 

1370.1     41.11      41.40     690.02 

44 

46 

1274.1      35.53      35.75    641.05 

1373.4     41.31      41.60     691.70 

46 

48 

1277.4      35.71      35.94    642.74 

1376.7     41.51      41.81     693.39 

48 

50 

1280.7      35.90      36.13    644.43 

1380.0     41.71      42.01     695.08 

50 

52 

1284.0      36.09      36.31    646.12 

1383.4     41.91      42.21     696.77 

52 

54 

1287.4      36.27      36.50    647.81 

1386.7     42.11      42.42     698.46 

54 

56 

1290.7      36.46      36.69    649.49 

1390.0     42.31       42.62     700.14 

56 

58 

1294.0      36.64      36.88    651.18 

1393.3     42.51      42.83     701.83 

58 

[60 

1297.3      36.83      37.07    652.87 

1396.6     42.71      43.03     703.53 

60 

6.— FUNCTIONS   OF   A   ONE-DEGREE   CURVE.    135 


14° 

15° 

L.  C.        M.         E.          T. 

L.  C.        M.         E.          T. 

0 

1396.6    42.71    43.03    ?03.53 

1495.9    49.02    49.44    754.85 

0 

2 

1399.9    42.92    43.23    705.23 

1499.2    49.24    49.66    756.05 

2 

4 

1403.2    43.12    43.44    706.92 

1502.5    49.46    49.89    757.74 

4 

6 

1406.5    43.33    43.65    708.62 

1505.8    49.68    50.11    759.44 

6 

8 

1409.8    43.53    43.86    710.31 

1509.1    49.90    50.34    761.13 

8 

10 

1413.1    48.74    44.07    712.01 

1512.4    50.12    50.56    762.88 

10 

12 

1416.5    43.94    44.28    713.71 

1515.7    50.34    50.78    764.53 

12 

14 

1419.8    44.15    44.49    715.40 

1519.0    50.56    51.01    766.22 

14 

16 

1423.1    44.35    44.70    717.10 

1522.3    50.78    51.23    767.92 

16 

18 

1426.4    44.56    44.91     718.79 

1525.6    51.00    51.46    769.61 

18 

20 

1429.7    44.77    45.12    720.49 

1528.9    51.22    51.68    771.31 

20 

22 

1433.0    44.98    45.33    722.20 

1532.2    51.44    51.90    773.01 

22 

24 

1436.3    45.19    45.54    723.89 

1535.5    51.67    52.13    774.70 

24 

26 

1439.6    45.40    45.76    725.59 

1538.8    51.89    52.36    776.40 

26 

28 

1442.9    45.61    45.97    727.28 

1542.1    52.12    52.59    778.09 

28 

30 

1446.2    45.82    46.18    728.97 

1545.4    52.34    52.82    779.79 

30 

32 

1449.6    46.03    46.40    730.66 

1548.7    52.57    53.05    781.49 

32 

34 

1452.9    46.24    46.61     732.35 

1552.0    52.79    53.28    783.19 

34 

36 

1456.2    46.45    46.82    734.05 

1555.3    53.02    53.51    784.89 

36 

38 

1459.5    46.66    47.04    735.74 

1558.6    53.24    53.74    786.59 

38 

40 

1462.8    46.87    47.25    737.43 

1561.9    53.47    53.97    788.29 

40 

42 

146G.1    47.08    47.46    739.12 

1505.2    53.69    54.20    789.99 

42 

44 

1469.4    47.30    47.68    740.81 

1568.5    53.92    54.44    791.69 

44 

46 

1472.7    47.51    47.90    742.51 

1571.8    54.15    54.67    793.39 

46 

48 

1476.  0    47.73    48.12    744.20 

1575.1    54.38    54.91     795.09 

48 

50 

1479.3    47.94    48.34    745.89 

1578.4    54.61    55.14    796.79 

50 

52 

1482.7    48.16    48.56    747.58 

1581.7    54.84    55.37    798.49 

52 

54 

1486.0    48.37    48.78    749.27 

1585.0    55.07    55  61    800.19 

54 

56 

1489.3    48.59    49.00    750.97 

1588.3    55.30    55.84    801.89 

56 

58 

1492.6    48.80    49.22    752.66 

1591.6    55.53    56.08    803.59 

58 

60 

1495.9    49.02    49.44    754.35 

1594.9    55.76    56.31    805.29 

60 

16° 

17° 

L  C.        M.         E.         T. 

L.  C.        M.         E.          T. 

0 

1594.9    55.76    56.31    805.29 

1693.9    62.94    63.64    856.35 

0 

2 

1598.2    55.99    56.54    806.99 

1697.2    63.18    63.89    858.05 

2 

4 

1601.5    56.23    56.78    808.64 

1700.5    63.43    64.15    859.76 

4 

6 

1604.8    56.46    57.02    810.39 

1703.8    63.68    64.40    861.46 

6 

8 

1608.1    56.70    57.26    812.09 

1707.1    63.93    64.66    863.16 

8 

10 

1611.4    56.93    57.50    813.79 

1710.4    64.18    64.91    864.87 

10 

12 

1614.7    57.17    57.74    815.49 

1713.7    64.42    65.16    866.57 

12 

14 

1618.0    57.40    57.98    817.19 

1716.9    64.67    65.42    868.27 

14 

16 

1621.3    57.64    58.22    818.89 

1720.2    64.92    65.67    869.98 

16 

18 

1624.6    57.87    58.46    820.59 

1723.5    65.17    65.93    871.68 

18 

20 

1627.9    58.11    58.70    822.29 

1726.8    65.42    66.18    873.38 

20 

22 

1631.2    58.34    58.94    823.99 

1730.1    65.67    66.43    875.09 

22 

24 

1634.5    58.58    59.19    825.69 

1733.4    65.93    66.69    876.79 

24 

26 

1637.8    58.82    59.43    827.39 

1736.7    66.18    66.95    878.49 

26 

28 

1641.1    59.06    59.68    829.09 

1740.0    66.44    67.21    880.20 

28 

30 

1644.4    59.30    59.92    830.79 

1743.3    66.69    67.47    881.90 

30 

32 

1647.7    59.54    60.16    832.49 

1746.6    66.94    67.72    883.61 

32 

84 

1651.0    59.78    60.41    834.20 

1749.9    67.20    67.98    885.32 

34 

36 

1654.3    60.02    60.65    835.90 

1753.2    67.45    68.24    887.02 

36 

38 

1657.6    60.26    60.90    837.61 

1756.5    67.71    68.50    888.73 

38 

40 

1660.9    60.50    61.14    839.31 

1759.8    67.96    68.76    890.44 

40 

42 

1664.2    60.74    61.39    841.01 

1763.1     68.21     69.03    892.15 

42 

44 

1667.5    60.99    61.64    842.72 

1766.3    68.47    69.29    893.86 

44 

46 

1670.8    61.23    61.89    844.42 

1769.6    68.73    69.56    895.56 

46 

48 

1674.1    61.48    62.14    846.13 

1772.9    68.99    69.82    897.27 

48 

50 

1677.4    61.72    62.39    847.83 

1776  2    69.25    70.09    898.98 

50 

52 

1680.7    61.96    62.64    849.53 

1779.5    69.50    70.36    900.69 

52 

54 

1684.0    62.21    62.89    851.24 

1782.8    69.76    70.62    902.40 

54 

56 

1687.3    62.45    63.14    852.94 

1786.1    70.02    70.89    904.10 

56 

58 

1690.6    62.70    63  39    854.65 

1789.4    70.28    71.17     905.81 

58 

60 

1693.0    62.94    63.64    856.35 

1792.7    70.54    71.42     907.52 

60 

136    6.— FUNCTIONS  OF  A  ONE-DEGREE   CURVE. 


18° 

19° 

/ 

L.  C.       M.          E.           T. 

L.  C.       M.          E.           T. 

0 

1792.7    70.54    71.42      907.52 

1891.5    78.58    79.65      958.86 

0 

2 

1796.0    70.80    71.69      909.23 

1894.8    78.86    79.94      960.57 

2 

4 

1799.3    71.06    71.96      910.94 

1898.1    79.13    80.22      962.30 

4 

6 

1802.6    71.33    72.23      912.65 

1901.3    79.41     80.51      964.00 

6 

8 

1805.9    71.59    72.50      914.36 

1904.6    79.68    80.79      965.72 

8 

10 

1809.2    71.85    72.77      916.07 

1907.9    79.96    81.08      967.43 

10 

12 

1812.5    72.12    73.04      917.78 

1911.2    80.24    81.37      969.15 

12 

14 

1815.7    72.38    73.31      919.49 

1914.5    80.51     81.65      970.86 

14 

16 

1819.0    72.64    73.58      921.20 

1917.8    80.79    81.94      972.58 

16 

18 

1822.3    72  91     73.85      922.91 

1921.0    81.07    82.22      974.29 

18 

20 

1825.6    73.17    74.12      924.63 

1924.3    81.35    82.51      976.01 

20 

22 

1828.9    73.43    74.39      926.34 

1927.6    81.63    82.80      977.72 

22 

24 

1832  2    73.70    74.67      928.05 

1930.9    81.91    83.09      979.44 

24 

26 

1835.5    73.97    74.94      929.76 

1934.2    82.20    8338      981.15 

26 

28 

1838.8    74.24    75.22      931.47 

1937.5    82.48    83.67      982.86 

28 

30 

1842.1    74.51    75.49      933.18 

1940.7    82.76    83.97      984.58 

30 

32 

1845.4    74.77    75.77      934.89 

1944.0    83.05    84.26      986.30 

32 

34 

1848.7    75.04    76.04      936.60 

1947.3    83.33    84.55      988.02 

34 

36 

1852.0    75.31    76.32      938.32 

1950.6    83.61    84.84      989.74 

36 

38 

1855.3    75.58    76.59      940.03 

1953.9    83.90    85.13      991.46 

38 

40 

1858.6    75.85    76.87      941.74 

1957.2    84.18    85.43      993.18 

40 

42 

1861.9    76.12    77.14      943.45 

19604    84.47    85.73      994.90 

42 

44 

1865.1    76.39    77.42      945.16 

1903.7    84.75    80.02      996.62 

44 

46 

1868.4    76.67    77.70      946.88 

1967.0    85.04    86  32      998.34 

46 

48 

1871.7    76.94    77.98      948.59 

1970.3    85.32    86.61    1000.0 

48 

50 

1875.0    77.21    78.26      950.30 

1973.6    85.61    86.91    1001.8 

50 

52 

1878.3    77.49    78.53      952.01 

1976.9    85.90    87.21     1003.5 

52 

54 

1881.6    77.76    78.81      953.72 

1980.1    86.19    87.50    1005.2 

54 

50 

1884.9    78.03    79.09      955.44 

1983.4    86.47    87.80    1006.9 

56 

58 

1888.2    78.31    79.37      957.15 

19S6.7    86.76    88.09    1008.6 

58 

60 

1891.5    78.58    79.  (',5      958  86 

1990.0    87.05    88.39    1010.4 

60 

20° 

21° 

L.  C.       M.         E.           T. 

L,  C.         M.          E.           T. 

/ 

0 

1990.0    87.05    88.39      1010.4 

2088.5      95.95      97.58    1062.0 

0 

2 

1993.3    87.34    88.69      1012.1 

2091.8      96.26      97.90    1063.7 

2 

4 

1996.6    87.  G3    88.99      1013.8 

2095.0      96.56      98.21     1065.4 

4 

6 

1999.8    87.92    89.29      1015.5 

2098.3      96.87      98.53    1067.2 

6 

8 

£003.1    88.21    89.59      1017.2 

2101.6      97.17      98.84    1068.9 

8 

10 

2006.4    88.50    89.89      1019.0 

2104.9      97.48      99.16    1070.6 

10 

12 

2009.7    88.79    90.19      1020.7 

2108.1      97.79      99.48    1072.4 

12 

14 

2013.0    89.08    90.49      1022  4 

2111.4      98.09      99.79    1074.1 

14 

16 

2016  3    89.37    90.79      1024.1 

2114.7      98.40     100.1      1075.8 

16 

18 

2019.5    89.66    91.09      1025.8 

2118.0      98.70     100.4      1077.5 

18 

20 

2022.8    89.96    91.40      1027.6 

2121.2      99.00     J00.7      1079.3 

20 

22 

2G26.1    90.25    91.71      1029.3 

2124.5      99.30    J01.1      1081.0 

22 

24 

2029-4    90.55    92.01      1031.0 

2127.8      99.60    J01.4      1082.7 

24 

26 

2032.7    90.85    92.32      1032.7 

2131.0      99.90    101.7      1084.4 

26 

28 

2036.0    91.15    92.  G2      1034.4 

2134.3    100.2      102.0      1086.2 

28 

30 

2039.2    91.45    92.93      1036  1 

2137.6    100.5      102.3      1087.9 

30 

32 

2042.5    91.74    93.24      1037.9 

2140.9    100.8      102.7      1089.6 

32 

34 

2045.8    92.04    93.54      1039.6 

2144.1    101.1      1030      1091.3 

34 

36 

2049.1    92.34    93.85      1041.3 

2147.4    101.4      103.3      1093.1 

36 

38 

2052.4    92.64    94.15      1043.0 

2150.7    101.7      103.6      1094.8 

88 

40 

2055.7    92.94    94.46      1044.8 

2154.0    102.1       104.0      1096.5 

40 

42 

2058.9    93.24    94.78      1046.5 

2157.2    102.4      104.3      1098.3 

42 

44 

2062.2    93.54    95.09      1048.2 

2160.5    102.7      104.6      1100.0 

44 

46 

2065.5    93.84    95.40      1049.9 

2163.8    103.0      104.9      1101.7 

46 

48 

2068.8    94.14    95.71       1051.7 

2167.1     103.3      105.3      1103.4 

48 

50 

2072.1     94.44    96.03      1053.4 

2170.3    103.6      105.6      1105.2 

50 

52 

2075.4    94.74    96.34      1055.1 

2173.6    103.9      105.9      1106.9 

52 

54 

2078.6    95.04    96.65      1056.8 

2176.9    104.2      106.3      1108.6 

54 

56 

2081.9    95.34    96  96      1058.6 

2180.1     104.5      106.6      1110.3 

56 

58 

2085.2    95.64    97  27      10G0.3 

2183.4    104.8      106.9      1112.1 

58 

60 

2088.5    95.95    97.58      1062.0 

2186.7    105.2      107.2      1118.8 

60 

6.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.      137 


22° 

23° 

L.  C.        M.         E.          T. 

L.  C.        M.          E.         T. 

0 

2186.7    103.2    107.2    1113.8 

2284.8    115.0    117.4    1165.8 

0 

2 

2190.0    105.6    107.6    1115.5 

2288.1     115.3    117.7    1167.5 

2 

4 

2193.2    105.9    107.9    1117.3 

2291.3    115.-7    118.1     1169.2 

4 

6 

2196.5    106.2    108.2    1119.0 

2294.6    116.0    118.4    1171.0 

6 

8 

2199.8    106.5    108.6    1120.7 

2297.8    116.4    118.8    1172.7 

8 

10 

2203.0    106.8    108.9    1122.4 

2301.1     116.7    119.1    1174.4 

10 

12 

2206.3    107.1    109.2    1124.2 

2304.4    117.0    119.5    1176.2 

12 

14 

2209.6    107.4    109.6    1125.9 

2307.6    117.4    119.8    1177.9 

14 

16 

2212.9    107.7    109.9    1127.6 

2310.9    117.7    120.2    1179.7 

16 

18 

2216.1     108.0    110.2    1129.4 

2314.1    118.1     120.5    1181.4 

18 

20 

2219.4    108.4    110.6    1131.1 

2317.4    118.4    120.9    1183.1 

20 

22 

2222.7    108.7    110.9    1132.8 

2320.7    118.7    121.2    1184.9 

22 

24 

2-225.9    109.0    111.2    1134.6 

2323.9    119.1     121  6    1186.6 

24 

26 

2229.2    109.4    111.6     1136.3 

2327.2    119.4     121.9    1188.4 

26 

28 

2232.5    109.7    111.9    1138.0 

2330.4    119.8    122.3    1190.1 

28 

30 

2235.7    110.0    112.3    1139.7 

2333.7    120.1    122.6    1191.8 

30 

32 

2239.0    110.4    112.6    1141.5 

2337.0    120.4    123.0    1193.6 

32 

34 

2242.3    110.7    112.9    1143.2 

2340.2    120.8    123.3    1195.3 

34 

36 

2245.6    111.0    113.3    1144.9 

2343.5    121.1     123.7    1197.1 

36 

38 

2248.8    111.4    113.6    1146.7 

2346.7    121.5    124.1    1198.8 

38 

40 

2252.1    111.7    113.9    1148.4 

23500    121.8    124.4    1200.5 

40 

42 

2255.4    112.0    114.3    1150.1 

2353.3    122.1     124.8    1202.3 

42 

44 

2258.6    112.3    114.6    1151.9 

2356.5    122.5    125.1    1204.0 

44 

46 

2261.9    112.7    115.0    1153.6 

2359.8    122.8    125.5    1205.8 

46 

48 

2265.2    113.0    115.3    1155.4 

2303.0    123.2    125.8    1207.5 

48 

50 

2268.4    113.3    115.7    1157.1 

2306.3    123.5    126.2    1209.2 

50 

52 

2271.7    113.7    116.0    1158.8 

2369.6    123.8    126.6    1211.0 

52 

54 

2275.0    114.0    116.3    1160.6 

2372.8    124.2    126.9    1212.7 

54 

56 

2278.3    114.3    116.7    1162.3 

2376.1     124.5    127.3    1214.5 

56 

58 

2281.5    114.7    117.0    1164.0 

2379.3    124.9    127.6    1216.2 

58 

60 

2284.8    115.0    117.4    1165.8 

2382.6    125.2    128.0    1218.0 

60 

24° 

25° 

/ 

L.  0.        M.         E.         T. 

L  C.        M.          E.         T. 

0 

23S2.6    125.2    128.0    1218.0 

2480.4    135.8    139.1     1270.3 

0 

2 

2385.9    125.5    128.4    1219.7 

2483.6    186.2    139.5    1272.0 

2 

4 

2389.1     125.9    128.7    1221.4 

2486.9    136.5    139.9    1273.8 

4 

6 

2392.4    126.2    129.1     1223.2 

2400.1    136.9    140.3    1275.5 

6 

8 

2395.6    126.6    129.5    1224.9 

2493.4    137.2    140.6    1277.3 

8 

10 

2398.9    126.9    129.8    1226.7 

2496.6    137.6    141.0    1279.0 

10 

12 

2402.2    127.3    130.2    1228.4 

2499.9    138.0    141.4    1280.8 

12 

14 

2405.4    127.6    130.6    1230.2 

2503.1     138.3    141.8    1282.5 

14 

Ifl 

2408.7    128.0    130.9    1231.9 

2506.4    138.7    142.2    1284.3 

16 

18 

2411.9    128.3    131.3    1233/6 

2509.6    139.0    142.5    1286.1 

18 

20 

2415.2    128.7    131.7    1235.4 

2512.9    139.4    142.9    1287.8 

20 

22 

2418.5    129.0    132.0    1237.1 

2516.1     139.8    143.3    1289.6 

22 

24 

2421.7    129.4    132.4    123S.9 

2519.4    140.1     143.7    1291.3 

24 

26 

2425.0    129.7    132.8    1240.6 

2522.6    140.5    144.1     1293.1 

26 

28 

2428.2    130.1     133.1    1242.4 

2525.9    140.8    144.5    1294.8 

28 

30 

2431.5    130.4    133.5    1244.1 

2529.1    141.2    144.9    1296.6 

30 

32 

2434.8    130.8    133.9    1215.8 

2532.4    141.6    145.3    1298.3 

32 

34 

2433.0    131.1     134.2    1247.6 

2535.6    142.0    145.6    1300.1 

34 

36 

2441.3    131.5    134.8    1249.  3 

2538.9    142.3    146.0    1301.8 

36 

38 

2444.5    131.8    135.0    1251.1 

2542.1     142.7    146.4    1303.6 

38 

40 

2447.8    132.2    135.4    1252.8 

2545.4    143.1     146  8    1305.3 

40 

42 

2451.1     132.6    135.7    1254.6 

2548.  d    143.5    147.2    1307.1 

42 

44 

2454.3    132.9    136.1    1256.3 

2551.9    143.8    147.6    1308.8 

44 

46 

2457.6    133.3    136.5    1258.1 

2555.1    144.2    148.0    1310.6 

46 

48 

2460.  S    133.6    136.9    1259.8 

2558.4    144.5    148.4    1312.4 

48 

50 

2464.1     134.0    137.2    1261.5 

2561.6    144.9    148.8    1314.1 

50 

52 

2467.4    134.4    137.6    1263.3 

2564.9    145.3    149.2    1315.9 

52 

54 

2470.6    134.7    138.0    1265.0 

2568.1    145.7    149.5    1317.6 

54 

56 

2473.9    135.1     138.4    1266.8 

2571.4    146.0    149.9    1319.4 

56 

58 

2477.1    135.4    138.7    1263.5 

2574.6    146.4    150.3    1321.1 

58 

60 

2480.4    135.8    139.1    1270.3 

2577.9    1468    150.7    1322.9 

60 

138      6.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


26° 

27* 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

I 

0 

2577.9  146.8  150.7  1322.9 

2675.3  158.3  162.8  1375.6 

0 

2 

2581.1  147.1  151.1  1324.6 

2678.5  158.6  163.2  1377.4 

2 

4 

2584.4  147.5  151.5  1326.4 

2681.8  159.0  163.7  1379.2 

4 

6 

2587.6  147.9  151.9  1328.1 

2685.0  159.4  164.1  1380.9 

6 

8 

2590.9  148.3  152.3  1329.9 

2688.2  159.8  164.5  1882.7 

8 

10 

2594.1  148.7  152.7  1331.6 

2691.5  160.2  164.9  1384.5 

10 

12 

2597.4  149.1  153.1  1333.4 

2694.7  160.6  165.3  1386.2 

12 

14 

2600.6  149.4  153.5  1335.2 

2698.0  161.0  165.7  1388.0 

14 

16 

2603.9  149.8  153.9  1336.9 

2701.2  161.4  166.1  1389.8 

16 

18 

2607.1  1502  154.3  1338.7 

2704.4  161.8  166.5  1391.5 

18 

20 

2610.4  1506  154.7  1340.4 

2707.7  162.2  167.0  1393.3 

20 

22 

2613.6  151  0  155.1  1342.2 

2710.9  162.6  167.4  1395.0 

22 

24 

2616.9  151.4  155.5  1343.9 

2714.1  163.0  167.8  1396.8 

24 

26 

2620.1  151.7  155.9  1345.7 

2717.4  163.4  168.2  1398.6 

26 

28 

2623.4  152.1  156.3  1347.4 

2720.6  168.8  168.6  1400.3 

28 

30 

2626.6  152.5  156.7  1349.2 

2723.8  164.2  169.1  1402.1 

30 

32 

2629.8  152.9  157.1  1351.0 

2727.1  164.6  169.5  1403.9 

32 

34 

2633.1  153.3  157.5  1352.7 

2730.3  165.0  169.9  1405.6 

34 

36 

2636.3  153.7  157.9  1354  5 

2733.6  165.4  170.3  1407.4 

36 

38 

2639.6  154.0  158.3  1356.2 

2736.8  165.8  170.8  1409.2 

38 

40 

2642.8  154.4  158.7  1358.0 

2740.0  166.2  171.2  1410.9 

40 

42 

2646.1  154.8  159.1  1359.8 

2743.3  166.6  171.6  1412.7 

42 

44 

2649.3  155.2  159.5  1361.5 

2746.5  167.0  172.0  1414.5 

44 

46 

2652.6  155.6  160.0  1363.3 

2749.7  167.4  172.5  1416.3 

46 

48 

2655.8  156.0  160.4  1365.1 

2753.0  167.8  172.9  1418.0 

48 

50 

2659.1  156.3  160.8  1366.8 

2756.2  168.2  173.3  1419.8 

50 

52 

2662.3  156.7  161.2  1368.6 

2759.5  168.6  173.7  1421.6 

52 

54 

2665.6  157.1  161.6  13~0.4 

2762.7  169.0  174.1  1423.3 

54 

56 

2668.8  1575  162.0  13~2.1 

2765.9  169.4  174.6  1425.1 

56 

58 

2672.1  157.9  162.4  1373.9 

2769.2  169.8  175.0  1426.9 

58 

60 

2675.3  158.3  162.8  1375.6 

2772.4  170.2  175.4  1428.6 

60 

28° 

29° 

1 

/ 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

2772.4  170.2  175.4  1128.6 

2869.4  182.5  188.5  1481.9 

0 

2 

2775.6  170.6   75.8  1430.4 

2872.6  182.9  189.0  1483.7 

2 

4 

2778.9  171.0   76.3  1432.2 

2875.8  183  3  189.4  1485.4 

4 

6 

2782.1  171.4   76.7  1434.0 

2879.1  183.7  189.9  1487.2 

6 

8 

2785.3  171.8   77.1  1435.7 

2882.3  18^.2  190.3  1489.0 

8 

10 

2788.6  172.2   77.6  1437.5 

2885.5  184.6  190.8  1490.8 

10 

12 

2791.8  172.6   78.0  1439.3 

2888.7  185.0  191.2  1492.6 

12 

14 

2795.0  173.0   78.4  1441.1 

2892.0  185.4  191.7  1494.3 

14 

16 

2798.3  173.4   78.9  1442.8 

2895.2  185.8  192.1  1496.1 

16 

18 

2801.5  173.8   79.3  1444.6 

2898.4  186.3  192.5  1497.9 

18 

20 

2804.7   74.3  179.7  1446.4 

2901.6  186.7  193.0  1499.7 

20 

22 

2808.0   74.7  180.2  1448.2 

2904.8  187.1  193.5  1501.5 

22 

24 

2811.2   75.1  180.6  1449.9 

2908.1  187.5  193.9  1503.2 

24 

26 

2814.4   75.5  181.0  1451.7 

2911.3  188.0  194.4  1505.0 

26 

28 

2817.7   75.9  181.5  1453.5 

2914.5  188.4  194.8  1506.8 

28 

30 

2820.9   76.3  181.9  1455.2 

2917.7  188.8  195.3  1508-6 

30 

32 

2824.1  176.7  182.3  1457.0 

2921.0  189.2  195.7  1510.4 

32 

34 

2827.4  177.1  182.8  1458.8 

2924.2  189.7  196.2  1512.1 

34 

36 

2830.6  177.5  183.2  1460.6 

2927.4  190.1  196.7  1513.9 

36 

38 

2833.8  177.9  J83.6  1462.3 

2930.6  190.5  197.1  1515.7 

38 

40 

2837.1  178.4  184.1  1464.1 

2933.9  190.9  197.6  1517.5 

40 

42 

2840  3  178.8  184.5  1465.  9 

2937.1  191.4  198.0  1519.3 

42 

44 

28435  179.2  185.0  H67.7 

2940.3  191.9  198.5  1521.0 

44 

46 

2846.8  179.6  185.4  1469.5 

2943.5  192.4  198.9  1522.8 

46 

48 

2850.0  1800  185.9  1471.2 

2946.8  192.8  199.4  1524.6 

48 

50 

2853.2  180.4  186.3  1473.0 

2950.0  193.2  199.8  1526.4 

50 

52 

2856.5  180.8  186.8  1474.8 

2953.2  193.6  200.3  1528.2 

52 

54 

2859.7  181.2  187.2  1476.6 

2956.4  194.0  200.8  1530.0 

54 

56 

2862.9  181.6  187.6  1478.3 

2959.6  194.4  201.2  1531.7 

56 

58 

2866.2  182.0  183.1  1480.1 

2962.9  194.8  201.7  1533.5 

58 

60 

2869.4  182  5  188.5  1481.9 

2966.1  195.2  202.1  1535  3 

60  1 

6.—  FUNCTIONS  OF  A  ONE-DEGREE  CURVE.     139 


30° 

31° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

2966.1  195.2  202.1  1535.3 

3062.6  208.4  216  3  1589.0 

0 

2 

2969.3  195.6  202.6  1537.1 

3065.8  208.8  216.8  1590.8 

2 

4 

2972.5  196.1  203.1  1538.9 

3069.0  209.3  217  2  1592.6 

4 

6 

2975.7  196.5  203.5  1540.7 

3072.2  209.7  217.7  1594.4 

6 

8 

2979.0  197.0  204.0  1542.5 

3075.4  210.2  218.2  1596.2 

8 

10 

2982.2  197.4  204.5  1544  3 

3078.6  210.6  218.7  1598.0 

10 

12 

2985.4  197.8  204.9  1546.0 

3081.8  211.1  219.2  1599.8 

12 

14 

2988.6  198.2  205.4  1547.8 

3085.0  211.5  219.6  1601.6 

14 

16 

2991.8  198.6  2059  1549.6 

3088.3  212.0  220.1  1603.4 

16 

18 

29950  199.1  2063  1551.4 

3091.5  212.4  220.6  1605.2 

18 

20 

2998.3  199  5  206.8  1553.2 

3094.7  212.9  221  1  1607.0 

20 

22 

3001.5  199.9  207.3  1555.0 

3097.9  213.3  221.6  1608.8 

22 

24 

3004.7  200.4  207.7  1556.8 

3101.1  213.8  222.1  1610.6 

24 

26 

3007.9  200.8  208.2  1558.6 

3104.3  214.2  222.6  1612.4 

26 

28 

3011  1  201.3  208.7  1560.4 

3107.5  214.7  223.0  1614.2 

28 

30 

3014.3  201.7  209.1  1562.2 

3110.7  215.1  223.5  1616.0 

30 

32 

30176  2021  209.6  1564.0 

3113.9  215.6  224.0  1617.8 

32 

34 

3020.8  202  6  210  1  1565.7 

3117.1  216.0  224.5  1619.6 

34 

36 

3024.0  203.0  210.5  1567.5 

3120.3  216.5  225.0  1621.4 

36 

38 

3027.2  203.5  211.0  1569.3 

3123.5  216.9  225.5  1623.2 

33 

40 

3030.4  203.9  211  5  1571.1 

3126.7  217.4  226.0  1625.0 

40 

42 

3033.6  204.3  212  0  1572.9 

3129.9  217.8  226.5  1626.8 

42 

44 

3036.9  204.8  212.4  1574.7 

3133.1  218.3  227.0  1628.6 

44 

46 

3040.1  205.2  212.9  1576.5 

3136.4  218.7  227.5  1630.5 

46 

48 

3043.3  205.7  213.4  1578.3 

3139.6  219.2  228.0  1632.3 

48 

50 

3046.5  206.1  213.9  1580.1 

3142.8  219.6  228.4  1634.1 

50 

52 

3049.7  206.5  214.4  1581.9 

3146.0  220.1  228.9  1635.9 

52 

54 

3052.9  207  0  214.8  1583.7 

3149.2  220.5  229.4  1637.7 

54 

56 

3056.2  207.4  215.3  1585.5 

3152.4  221.0  2.9.9  1639.5 

56 

58 

3059.4  207.9  215.8  1587.2 

3155.6  221.5  230.4  1641.3 

58 

60 

3062  6  208.4  216  3  1589.0 

3158.8  222.0  230.9  1643.1 

60 

32° 

33° 

L.  C.   M.    E.    T. 

L.  C    M.    E.    T. 

/ 

0 

3158.8  222.0  230.9  1643.1 

3254.0  236.0  246.1  1697  3 

0 

2 

3162.0  222  5  231.4  1644  9 

3258.1  236.4  246.6  1699.1 

2 

4 

3165.2  222.9  231.9  1646.7 

3261.3  236.9  247.1  17009 

4 

6 

3168.4  223.4  232.4  1648.5 

3264.5  237.4  247.7  1702.7 

6 

8 

3171.6  223.8  232.9  1650.3 

3267  7  237.9  248.2  1704.5 

8 

10 

3174.8  224.3  233.4  1652.1 

3270.8  238.4  248.7  1706.4 

10 

12 

3178.0  224  8  233.9  1653.9 

3274  0  238.9  249.2  1708  .-2 

12 

14 

3181.2  225.2  234.4  1655  7 

3277.2  2393  249.7  1710.0 

14 

16 

3184.4  225.7  234.9  1657.5 

3280.4  239.8  250.2  1711.8 

16 

18 

3187.6  226.1  235.4  1659.3 

3283.6  240.3  250.8  1713.6 

18 

20 

3190.8  226.6  2359  1661.1 

3286.8  240.8  251.3  1715.5 

20 

22 

3194.0  227.1  236.4  1662.9 

3290  0  241.2  251.8  1717.3 

22 

24 

3197  2  227.5  236  9  1664.7 

3293.2  241.7  252.3  1719.1 

24 

26 

3200.4  228.0  237.4  1666  5 

3296.4  242.2  252.9  1720.9 

26 

28 

3203.6  228.4  2379  1668.3 

3299.6  242.7  253.4  1722.7 

28 

30 

3206.8  228  9  238.4  1670.1 

3302.7  243  2  253.9  1724.6 

30 

32 

3210.0  229.4  239.0  1671.9 

3305  9  243.6  254.4  1726.4 

32 

34 

3213.2  229  8  239.5  1673  7 

33091  244.1  255.0  1728.2 

34 

36 

3216  5  230.3  240.0  1675.5 

3312  3  244.6  255.5  1730.0 

36 

88 

8219.7  230.7  240.5  1677.4 

3315  5  245.1  256.0  1731.8 

38 

40 

32229  231  2  241.0  1679.2 

3318.7  245.6  256.5  1733.6 

40 

42 

3226  1  231.7  241.5  1681.0 

3321.9  246.0  257.1  1735.5 

42 

44 

3629.3  232.2  242.0  1682  8 

3325.1  246.5  257.6  1737.3 

44 

46 

3232.5  232.6  242.5  1684.6 

3328.3  247.0  258.1  1739.1 

46 

48 

3235.7  233.1  243.0  16864 

3331.5  247  5  258.6  1740.9 

48 

50 

3238.9  233.5  243.5  16882 

3334.6  248.0  259  2  1742.7 

50 

52 

3242.1  234.0  244.1  1690.0 

3337.8  248.4  259.7  1744.6 

52 

54 

3245  3  234.5  244.6  1691.8 

3341.0  248.9  2602  1746.4 

54 

66 

3248.5  235  0  245.1  1693.7 

3344.8  249.4  260.8  1748.2 

56 

58 

3251.7  235.5  245.6  1695  5 

3347.4  249.9  261.3  1750,0 

53 

60 

3254.9  236.0  246  1  1697.3 

3350-6  250.4  261  8  1751,8 

An 

140      6.— FUNCTIONS   OF   A   ONE-DEGREE   CURVE 


34° 

85° 

L.  C.        M.         E.         T. 

L.  0.        M.         E.         T. 

, 

0 

3350.6    250.4    261.8    1751.8 

3446.1    265.2    278.1     1806.7 

0 

2 

3353.8    250.8    262.3    1753.7 

3449.3    265.7    278.6    1808.5 

2 

4 

3357,0    251.2    262.9    1755.5 

3452.5    266.2    279.2    1810.3 

4 

6 

3360.1    251.7    263.4    1757.3 

3455.6    266.7    279.7    1812.2 

6 

8 

3363.3    252.2    2U4.0    1759.1 

3458.8    267.2    280.3    1814.0 

8 

10 

3366.5    252.7    264.5    1761.0 

3462.0    267.7    280.8    1815.8 

10 

12 

3369.7    253.2    265.0    1762.8 

3465.2    268.2    281.4    1817.7 

12 

14 

3372.9    253.7    265.6    1764.6 

3468.3    268.7    281.9    1819.5 

14 

16 

3376.1    254.2    266.1     1766.4 

3471.5    269.2    282.5    1821.3 

16 

18 

3379.2    254.7    266.7    1768.3 

3474.7    269.7    283.0    1823.2 

18 

20' 

3382.4    255.2    267.2    1770.1 

34V7.9    270.2    283.6    1825.0 

20 

22 

3385.6    255.7    267.7    1771.9 

3481.0    270.7    284.2    1820.8 

22 

24 

3388.8    256.2    268.3    1773.7 

3484.2    271.2    2S4.7    1828.7 

24 

26 

3392.0    256.7    268.8    1775.6 

3487.4    271.7    285.3    1830.5 

26 

28 

3395.2    257.2    269.3    1777.4 

3490.6    272.2    285.9    1832.3 

28 

30 

3398.3    257.7    269.9    1779.2 

3493.7    272.7    286.4    1834.2 

30 

83 

3401.5    258.2    270.4    1781.0 

3496.9    273.2    287.0    1830.0 

32 

34 

3404.7    258.7    271.0    1782.9 

3500.1    273.  Y    287.5    1837.8 

34 

36 

3407.9    2592    271.5    1784.7 

3503.3    274.2    288.1     1839.7 

8tf 

38 

3411.1    259.7    272.0    1786.5 

3506.5    274.7    288.7    1841.5 

38 

40 

3414.3    260.2    272.6    1788.4 

3509.6    275.2    289.2    1843.4 

40 

42 

3417.4    260.7    273.1     1790.2 

3512.8    275.7    289.8    1845.2 

42 

44 

3420.6    261.2    273.7    1792.0 

3516.0    276.2    290.4    1847.1 

44 

46 

3423.8    261.7    274.2    1793.9 

3519.2    -276.7    290.9    1848.9 

46 

48 

3427.0    262.2    274.8    1795.7 

3522.3    277.2    291.5    1850.7 

48 

50 

3430.2    262.7    275.3    1797.5 

3525.5    277.7    292.0    185;!.  6 

50 

52 

3433.4    263.2    275.9    1799.3 

3528.7    278.2    292.6    1854.4 

52 

54 

3436.5    263.7    276.4    1801.2 

3531.9    278.7    293.2    1856.3 

54 

56 

3439.7    264.2    277.0    1803.0 

3535.0    279.2    293.7    1858.1 

56 

58 

3442.9    264.7    277.5    1804.8 

3538.2    279.8    294.3    1859.9 

58 

00 

3446  1     265.2    278.1     1806.7 

3541.4    280.4    294.9    1801.8 

60 

36° 

37° 

L.  C.        M.         E.          T. 

L.  C.        M.         E.          T. 

0 

3541.4    280.4    294.9    1861.8 

3636.3    296.1    312.3    1917.3 

0 

2 

3544.6    280  9    295.4    1863.6 

3639.5    296.6    312.8    1919.1 

2 

4 

3547.7    281.4    296.0    1865.5 

3642  6    297  1    313.4    1921.0 

4 

6 

3550.9    281.9    296.6    1867.3 

3645.8    297.7    314.0    1922.8 

6 

8 

3554.0    282.  5    297.2    3869.2 

3648.9    298.2    314.6    1924.7 

8 

10 

3557.2    2830    297.7    1871.0 

3652.1    298.7    315.2    1926.5 

10 

12 

3560.4    283.5    298.3    1872.9 

3655.2    299.3    315.8    1928.4 

12 

14 

3563  5    284.0    298.9    1874.7 

3658.4    299.8    316.4    1930  2 

14 

16 

3566.7    284.6    299.5    1876.5 

3661.6    300.3    317.0    1932.1 

16 

18 

3569.9    285.1    300.0    1878.4 

3064.7    300.9    317.5    1933.9 

18 

20 

3573.0    285.6    300.6    1880.2 

3G67.9    301.4    318.1    1935.8 

20 

22 

3576.2    286.1     301.2    1882.1 

3671.0    301.9    318.7    1937.6 

22 

24 

3579.4    286.7    301.8    1883.9 

3674.2    302.5    319.3    1939.5 

24 

26 

3582.5    2S7.2    302.3    1885.8 

3677.3    303.0    319.9    1941.3 

26 

28 

3585.7    287.7    302.9    1887.6 

3680.5    303.5    320.5    1943.2 

28 

30 

3588.8    288.2    303.5    1889.5 

3683.6    304.1    321.1     1945.0 

30 

32 

3592.0    288.8    304.1     1891.3 

3686.8    304.6    321.7    1946.9 

32 

34 

3595.2    289.3    304.6    1893.2 

3690.0    305.1    322.3    1948.8 

34 

36 

3598.3    289  8    305.2    1895.0 

3693.1    305.7    322.9    1950.6 

36 

38 

3601.5    290.3    305.8    1896.9 

3696.3    306.2    323.5    1952.5 

33 

40 

3604.7    290.9    306.4    1898.7 

3699.4    306.7    324.2    1954.4 

40 

42 

3607.8    291.4    307.0    1900.6 

3702.6    307.3    324.8    1956.2 

42 

44 

3611.0    291.9    307.5    1902.4 

3705.7    307.8    325.4    1958.1 

44 

46 

3614.1     292.4    308.1     1904.3 

3708.9    308.3    326.0    1960.0 

46 

48 

3617.3    293.0    308.7    1906.1 

3712.1    308.9    326.6    1961.8 

48 

50 

3620.5    293.5    309.3    1908.0 

3715.2    309.4    327  2    1963.V 

50 

52 

3623  6    294.0    309.9    1909.8 

3718.4    309.9    327.8    1965.5 

52 

54 

3626.8    294.5    310.5    1911.7 

37'21.5    310.5    328.4     1967.4 

54 

56 

3630.0    295.1     311.1     1913.5 

3724.7    311.0    329.0    1969.3 

56 

58 

3633.1     295.6    311.7    1915.4 

3727.8    311.6    329.6    1971.1 

58 

60 

3636.3    296.1    312.3    1917.3 

3731.0    312.2    330.2    1973.0 

60 

6.  —FUNCTIONS  OF  A  ONE-DEGREE  CURVE.     141 


38° 

39° 

L.  C.    M.   E.     T. 

L.  G.   M    E.    T. 

0 

3731.0  312.2  330.2  19730 

3825.5  328-7  348.7  2029.1 

0 

2 

3734.1  312.7  330.8  1974.9 

3828.6  329.2  349  3  2031.0 

2 

4 

3737.3  313,3  331.4  1976-7 

3831.8  329.8  349.9  20329 

4 

6 

3740.4  313.8  332.0  1978-6 

3834.9  330-3  350.8  2034.7 

6 

8 

3743.6  314.4  332.6  1980.5 

3838.0  330.9  351.2  2036.6 

8 

10 

3746.7  314.9  333.2  1982.3 

3841.2  331  5  351.8  2038.5 

10 

12 

3749.9  315.5  333.8  1984.2 

3844  3  332.0  352.4  2040  4 

12 

14 

37530  3160  334-5  1986.1 

3847.4  332.6  353.1  2042  3 

14 

16 

37562  316.6  335.1  1987.9 

3850.5  333.2  353.7  2044.1 

16 

18 

3759.3  317.1  335  7  1989.8 

3853  7  333  7  354.3  2046.0 

18 

20 

3762  5  317  7  336  3  1991.7 

3856.8  334.3  354.9  2047.9 

20 

22 

3765.6  318  2  336.9  1993-6 

3860  0  334.9  355.6  2049.8 

22 

24 

3768  8  318.8  337.5  1995.4 

3863.1  335.4  356.2  2051.7 

24 

26 

3771.9  319.3  338.1  1997.3 

3866.2  336-0  356.9  2053.5 

26 

28 

3775.1  319.9  338.7  1999.2 

3869.4  336.6  357  5  2055.4 

28 

30 

3778.2  3204  339.4  2001.0 

3872.5  337.1  358.1  2057-3 

30 

32 

3781.4  321.0  340.0  2002.9 

3875  6  337  7  358.8  2059.2 

32 

34 

3784  5  321.5  340-6  2004.8 

3878.8  338.3  359  4  2061.1 

34 

36 

3787.7  322.1  341.2  2006.6 

3881.9  338.8  360.1  2063.0 

36 

38 

37308  322.6  341.8  2008.5 

3885.0  339  4  360.7  2064.8 

38 

40 

3794.0  3'.'3  2  342.4  2010  4 

3888.2  340.0  361.3  2066.7 

40 

42 

3797  1  323.7  343.1  2012.3 

3891.3  340.5  362.0  2068  6 

42 

44 

3800.3  3','4.3  343.7  2014.1 

3894.4  341  1  362.6  20^0.5 

44 

46 

3803.4  324.8  344.3  2016  0 

3897.6  341.7  363.3  2072.4 

46 

48 

3806.6  325.4  344.9  2017.9 

3900  7  342  2  363.9  2074  2 

48 

50 

3809.7  325  9  345.6  2019.7 

3903.8  342.8  364.5  2076  1 

50 

52 

3812.9.  326.5  346.2  2021.6 

3907.0  343  4  365.2  2078.0 

52 

54 

3816.0  327.0  346.8  2023.5 

3910.1  343.9  365.8  2079  9 

54 

56 

3819.2  3-27.6  347.4  2025  4 

3913.2  344.5  366.5  2081.8 

56 

68 

3822.3  323.1  3481  2027.2 

3916.4  345  1  367.1  2083.7 

58 

60 

3825.5  328.7  34S.  7  '  2029.1 

3919  5  345.6  3(57.7  2085-5 

60 

40° 

41° 

L.  C.   M.    E.     T. 

L.  C.   M.    E.    T. 

0 

3919.5  345.6  367.7  2085.5 

4013.4  3G2.9  387.4  2142.3 

0 

2 

3922.6  346.1  368.4  2087.4 

4016.5  C63.4  388.1  2144.2 

2 

4 

3925.8  346  7  369  0  2089.3 

4019.6  364.0  388.8  2146.1 

4 

6 

3928.9  347.2  369.7  2091.2 

4022.7  36J.5  389.4  2148.0 

6 

8 

3932  0  347  8  370.3  20^3.1 

4025.9  365.1  390.1  2149.9 

8 

10 

3935.1  348.4  371.0  2095  0 

4029.0  365.6  390.7  2151.9 

10 

12 

3938.3  348.9  371  6  2096  9 

4032.1  366.2  391.4  2158.8 

12 

14 

3941.4  349.5  372.3  2098.8 

4035.2  366.  8  302.1  2155.7 

14 

16 

3944.5  350.1  372.9  2i00.7 

4038.3  307.4  392.7  2157.6 

16 

18 

3947.7  350.7  373.6  21026 

4041.4  368.0  393.4  2159.5 

18 

20 

3950.8  351.3  374.3  2104.5 

4044.6  3G8.6  394.1  2161.4 

20 

22 

3953.9  351  8  374.9  2106.3 

4047.7  369.2  394.7  2163.3 

22 

24 

3957.1  352.4  375.6  2108.2 

4050.8  369.8  395.4  2165.2 

24 

26 

3960.2  353.0  376.2  2110.1 

4053.9  370.4  396.1  2167.1 

26 

28 

3963.3  353  6  376.9  21120 

4057.0  371.0  396.8  2169.0 

28 

30 

3966.4  354.2  377.5  2113  9 

4060.1  371.6  397.5  21709 

30 

32 

3969.6  354.7  378.2  2115.8 

4063.3  372.2  398.1  2172.8 

32 

34 

3972.7  355.3  378.8  21177 

4066.4  372.8  309.8  2174.7 

34 

81 

3975.8  355.9  379.5  2119  6 

4069.5  373.4  399.5  2176.6 

36 

38 

3979.0  356.5  380  1  2121.5 

4072.6  374.0  400.2  2178.5 

38 

40 

3982.1  357.1  380.8  2123.4 

4075.7  374.6  400.9  2180.4 

40 

42 

3985.2  357  6  381.4  2125  3 

4078.8  375.2  401.5  2182.4 

42 

44 

3988.4  358.2  382.1  2127.2 

4082.0  375.8  402.2  2184.8 

44 

46 

3991  5  358  8  882.8  2129.1 

4085.1  376.4  402.9  2186.2 

46 

48 

3994.6  359.4  383.4  2131.0 

4088.2  377.0  403.6  2188.1 

48 

50 

3997  7  860  0  384.1  2132.9 

4091.3  377.6  404.3  2190.0 

50 

52 

4000  9  360  5  384,8  2134.7 

4094.4  378.2  404.9  2191.9 

52 

54 

4004.0  361  1  385.4  2136.6 

4097.5  378.8  405.6  2193.8 

54 

56 

40071  361.7  386.1  2138.5 

4100.7  379.4  406.3  2195.7 

56 

58 

4010.3  362.3  386  8  2140  4 

4103.8  380.0  407.0  2197.6 

58 

60 

4013  4  3t52.9  387.4  2142.3 

4106.9  380.6  407.7  2199.5 

60 

142     6.— FUNCTIONS  OF  A   ONE-DEGREE  CURVE. 


42° 

43° 

/ 

L.  C.    M.   E.     T. 

L.  C.   M.   E.     T. 

/ 

0 

4106.9  380.6  407.7  2199.5 

4200.1  398.7  428.6  2257.1 

0 

2 

4110.0  381.2  408.3  2201.4 

4203.2  399.3  429.3  2259.0 

2 

4 

4113.1  381.8  409.0  2203.3 

4206.3  399.9  430.0  2261.0 

4 

6 

4116.2  382.4  409.7  2205.3 

4209.4  400.5  430.7  2262.9 

6 

8 

4119.3  383.0  410.4  2207.2 

4212.5  401.1  431.4  2264.8 

8 

10 

4122.4  383.6  411.1  2209.1 

4215.6  401.7  432.1  2266.7 

10 

12 

4125.5  384.2  411.8  2211.0 

4218.7  402.4  432.8  2268.7 

12 

14 

4128.6  384.8  412.5  2212.9 

4221.8  403.0  433.5  2270.6 

14 

16 

4131.8  385  4  413.2  2214.9 

4224.9  403.6  434.2  2272.5 

16 

18 

4134.9  38b  0  413.9  2216.8 

4228.0  404.2  434.9  2274.5 

18 

20 

4138.0  386  3  414.6  2218.7 

4231.1  404.8  435.6  2276.4 

20 

22 

4141.1  387.2  415.3  2220.6 

4234.2  405.4  430.  3  2278.3 

22 

24 

4144.2  387.8  416.0  2222.5 

4237.3  406.1  437.0  2280.2 

24 

26 

4147  3  388.4  416.6  2224.4 

4240.4  406.7  437.8  2282.2 

26 

28 

4150.4  389.0  417.3  2226.4 

4243.5  407.3  438.5  2284.1 

28 

30 

4153.5  389.6  418.0  2228.3 

4246.5  407.9  439.2  2286.0 

30 

32 

4156.6  390.2  418.7  2230.2 

4249.6  408.5  439.9  2288.0 

32 

34 

4159.7  390.8  419.4  223?.  1 

4252.7  409.1  440.6  2289.9 

34 

36 

4162.8  391.4  420.1  2234.0 

4255.8  409.8  441.4  2291.8 

36 

38 

4165.9  392.0  420.8  2236.0 

4258.9  410.4  442.1  2293.8 

38 

40 

4169.0  392.6  421.5  2237.9 

4262.0  411.0  442.8  2295.7 

40 

42 

4172.1  393.2  422.2  2239.8 

4265.1  411.6  443.5  2297.7 

42 

44 

4175.2  393.8  422.9  2241.7 

4268.2  412.2  444.2  2299.6 

44 

46 

4178.4  394.4  423.6  2243.6 

4271.3  412.8  445.0  2301.5 

46 

48 

4181.5  395.0  424.3  2245.6 

4274.4  413.5  445.7  2303.5 

48 

50 

4184.6  395.6  425.0  2247.5 

4277.5  414.1  446.4  2305.4 

50 

52 

4187.7  396.2  425.7  2249.4 

4280.6  414.7  447.1  2307.3 

52 

54 

4190.8  396.8  426.4  2251.3 

4283.7  415.3  447.8  2309.3 

54 

56 

4193.9  397.4  427.1  2253.3 

4286.8  415.9  448.6  2311.2 

56 

58 

4197.0  398.0  427.8  2255.2 

4289.9  416.5  449.3  2313.1 

58 

60 

4200.1  398.7  428.6  2257.1 

4293.0  417.2  450.0  2315.1 

60 

44° 

45° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

4293.0  417.2  450.0  2315.1 

4385.5  436.2  472.1  2373.4 

0 

2 

4296  1  417.8  450.7  2317.0 

4388.6  436.8  472.9  2375.4 

2 

4 

4299.2  418.4  451.5  2319.0 

4391.7  437.5  473.6  2377.3 

4 

6 

4302.2  419.1  452.2  2320.9 

4394.7  438.1  474.4  2379.3 

6 

8 

4305.3  419.7  4529  2322.8 

4397.8  438.8  475.1  2381.2 

8 

10 

4308.4  420.3  453.7  2324.8 

4400.9  439.4  475.9  2383.2 

10 

12 

4311.5  421.0  454.4  2326.7 

4404.0  440.0  4766  2385.2 

12 

14 

4314.6  421.6  455.1  2328.7 

4407.0  440.7  477.4  2387.1 

14 

16 

4317.7  422.2  455.9  2330.6 

4410.1  441.3  478.1  2389.1 

16 

18 

4320.7  422.9  456.6  2332.6 

4413.2  442.0  478.9  2391.0 

18 

20 

4323.8  423  5  457.3  2334.5 

4416.3  442.6  479.6  2393.0 

20 

22 

4326.9  424.1  458.1  2336.4 

4419.3  443.2  480.4  2394.9 

22 

21 

4330.0  424  8  458.8  2338.4 

4422.4  443.9  481.1  2396.9 

24 

26 

4333.1  425.4  459.5  23403 

4425.5  444.5  481.9  2398.8 

26 

28 

4336.2  426.0  460.3  2342.3 

4428.6  445.2  482.6  2400.8 

28 

30 

4339.2  426.7  461  0  2344.2 

4431.6  445.8  483.4  2402.8 

30 

32 

4342.3  427.3  461.7  2346.1 

443*.  7  446.4  484.2  2404.7 

32 

34 

4345.4  427  9  462.5  2348.1 

4437.8  447.1  484.9  2406.7 

34 

36 

4348.5  428  6  463.2  2350.0 

4440.9  447.7  485.7  2408.6 

36 

38 

4351.6  429.2  463.9  2352.0 

4444.0  448.3  486.5  2410.6 

3u 

40 

4354.7  429.8  464.7  2353.9 

4447.0  448.9  487.2  2412.6 

40 

42 

4357.7  430.5  465.4  2355.9 

4450.1  449.5  488.0  2414.5 

42 

44 

4360.8  431.1  466.2  2357.8 

4453.2  450.2  488.7  2416.5 

44 

46 

4363.9  431.7  466.9  2359.8 

4456.3  450.8  489.5  2418.5 

46 

48 

4367.0  432.4  467.7  2361.7 

4459.3  451.5  490.  3  2420.4 

48 

50 

4370.1  433.0  4684  2363.7 

4462.4  452.1  491.0  2422.4 

50 

52 

4373.2  433.6  469.1  2365.6 

4465.5  452.7  491.8  2424.4 

52 

54 

43762  434.3  469.9  2367.6 

4468.6  453.4  492.5  2426.3 

54 

56 

43^9.3  434.9  470.  6  2369.5 

4471.6  454.1  493.3  2428.3 

56 

5b 

4382.4  435.6  471.4  23^1.5 

4474.7  454.8  494.1  2430.2 

58 

60 

4385.5  436.2  472.1  2373.4 

4477.8  455.5  494.8  2432.2 

60 

6  —FUNCTIONS  OF  A  ONE-DEGREE  CURVE.      143 


46° 

47° 

L.  C.        M.         E.         T. 

L.  C.       M.         E.          T. 

/ 

0 

4477.8    455.5    494.8    2432.2 

4569.7    475.2    518.3    2491.5 

"~0~ 

2 

4480.9    456.1    495.6    2434.2 

4572.7    475.9    519.0    2493.4 

2 

4 

4483.9    456.8    496.5    2436.1 

4575.8    476.5    519.8    2495.4 

4 

6 

4487.0    457.4    497.2    2438.1 

4578.8    477.2    520.6    2497.4 

6 

8 

4490.0    458.1    497.9    2440.1 

4581.9    477.8    521.4    2499.4 

8 

10 

4493.1    458.7    498.7    2442.1 

4584.9    478.5    522.2    2501.4 

10 

12 

4496.2    459.4    499.5    2444.0 

4588.0    479.2    523.0    2508.4 

12 

14 

4499.2    460.0    500.3    2446.0 

4591.0    479.8    523.8    2505.4 

14 

16 

4502.3    460.7    501.0    2448.0 

45P4.1    480.5    524.6    2507.8 

16 

18 

4505.4    461.3    501.8    2449.9 

4597.1    481.1    525.4    2509.8 

18 

20 

4508.4    462.0    502.6    2451.9 

4600.2    481.7    526.2    2511.3 

20 

22 

4511.5    462.7    503.4    2453.9 

4603.2    482.3    527.0    2513.3 

22 

24 

4514.6    463.3    504.1    2455.9 

4606.3    483.0    527.8    2515.3 

24 

26 

4517.6    464.0    504.9    2457.8 

4609.3    483.7    528.6    2517.3 

26 

28 

4520.7    464.6    505.7    2459.8 

4612.4    484.8    529.4    2519.3 

28 

80 

4523.7    465.3    506.5    2461.8 

4615.4    485.0    530.2    2521.2 

30 

32 

4526.3    466.0    507.3    2463.  S 

4618.5    485.7    531.0    2523.2 

32 

34 

4529.9    466.6    508.0    2465.7 

4621.5    486.3    531.8    2525.2 

34 

36 

4532.9    467.3    508.8    2467.7 

4624.6    487.0    532.6    2527.2 

86 

88 

4536.0    467.9    509.6    2469.7 

4627.6    487.7    533.4    2529.2 

38 

40 

4539.1    468.6    510.4    2471.7 

4630.7    488.4    534.2    2531.2 

40 

42 

4542.1    469.3    511.1    2473.6 

4633.7    489.1     535.0    2533.2 

42 

44 

4545.2    469.9    511.9    2475.6 

4G36.8    489.8    535.8    2535.2 

44 

46 

4548.2    470.6    512.7    2477.6 

4639.8    490.5    536.6    2537.2 

46 

48 

4551.3    471.2    513.5    2479.6 

4642.9    491.2    537.4    2539.2 

48 

50 

4554.4    471.9    514.3    2481.6 

4645.9    491.9    538.2    2541.2 

50 

52 

4557.4    472.6    515.1     2483.5 

4649.0    492.6    539.0    2543.1 

52 

54 

4560.5    473.2    515.9    2485.5 

4652.0    493.3    539.8    2545.1 

54 

56 

4563.6'  473.9    516.7    2487.5 

4655.1     494.0    540.6    2547.1 

56 

58 

4566.6    474.5    517.5    2489.5 

4658.1     494.7    541.4    2549.1 

58 

60 

4569.7    475.2    518.3    2491.5 

4661.2    495.4    542.3    2551.1 

60 

48° 

49° 

/ 

L.  C.       M.         E.          T. 

L.  C.       M.         E.          T. 

0 

4661.2    495.4    542.3    2551.1 

4752.3    515.9    567.0    2611.3 

0 

2 

46C4.2    496.0    543.1    2553.1 

4755.3    516.5    567.8    2613.3 

2 

4 

4667.3    496.7    543.9    2555.1 

4758.4    517.2    568.7    2615.3 

4 

6 

4670.3    497.4    544.7    2557.1 

4761.4    517.9    569.5    2617.3 

6 

8 

4673.3    498.1     545.5    2559.1 

4764.4    518.6    570.3    2619.3 

8 

10 

4676.4    498.8    546.4    2561.1 

4767.4    519.3    571.2    2621.4 

10 

12 

4679.4    499.4    547.2    2563.1 

4770.5    520.0    572.0    2623.4 

12 

14 

4682.5    500.1     548.0    2565.1 

4773.5    520.7    572.8    2625.4 

14 

16 

4685.5    500.8    548.8    2567.1 

4776.5    521.4    573.7    2627.4 

16 

18 

4688.5    501.5    549.6    2569.1 

4779.6    522.1    574.5    2629.4 

18 

20 

4691.6    502.2    550.5    2571.1 

4782.6    522.8    575.3    2631.4 

20 

22 

4694.6    502.8    551.3    2573.1 

4785.6    523.5    576.2    2633.5 

22 

24 

4697.6    503.5    552.1    2575.1 

4788.7    524.2    577.0    2635.5 

24 

26 

4700.7    504.2    552.9    2577.1 

4791.7    524.9    677.9    2637.5 

26 

28 

4703.7    504.9    553.7    2579.1 

4791.7    525.6    578.7    2639.5 

28 

30 

4706.7    505.6    554.6    2581.1 

4797.7    526.3    579.6    2641.5 

30 

32 

4709.8    506.2    555.4    2583.1 

4800.8    527.0    580.4    2643.5 

32 

34 

4712.8    506.9    556.2    2585.1 

4803.8    527.7    581.3    2645.6 

34 

36 

4715.9    507.6    557.0    2587.2 

4806.8    528.4    582.1    2647.6 

36 

38 

4718.9    508.3    557.8    2589.2 

4809.9    529.1    583.0    2649.6 

38 

40 

4721.9    509.0    558.7    2591.2 

4812.9    529.8    583.8    2651.6 

40 

42 

47-25.0    509.6    559.5    2593.2 

4815.9    530.5    584.7    2653.7 

42 

44 

4728.0    510.3    560.3    2595.2 

4819.0    531.2    585.5    2655.7 

44 

46 

4731.0    511.0    561.2    2597.2 

4822.0    531.9    586.4    2657.7 

46 

48 

4734.1     511.7    562.0    2599.2 

4825.0    532.6    587.2    2659.7 

48 

50 

4737.1     512.4    562.8    2601.2 

4828.0    533.3    588.1    2661.8 

50 

52 

4740.2    513.1    563.7    2603.2 

4881.1    534.0    588.9    2663.8 

52 

54 

4743.2    513.8    564.5    2605.2 

4834.1    534.7    589.8    2665.8 

54 

56 

4746.2    514.5    565.3    2607.2 

4837.1    535.4    590.6    2667.8 

56 

58 

4749.3    515.2    566.2    2609.3 

4840.2    536.1    591.5    2669.9 

58 

60 

4752.3    515.9    567.0    2611.3 

4843.2    586.8    592.4    2671.9 

60 

144     6. -FUNCTIONS   OF   A   ONE-DEGREE   CURVE. 


60° 

61° 

L.  C.   M.   E.    T. 

L.  C.    M.    E.    T. 

/ 

0 

4843.2  536.8  592.4  2671.  9 

4933  6  558.2  618.5  2733.0 

0 

2 

4846.2  537.5  593.2  2673  9 

4936.6  558.9  619.3  2735.1 

2 

4 

4849.2  538.2  594.1  2676.0 

4939.6  559.7  620.2  2737.1 

4 

6 

4852.2  538.9  594.9  2678.0 

4942.6  560.4  621.1  2739.2 

6 

8 

4855.2  539.6  595.8  2680.0 

4945.6  561.1  622.0  2741.2 

8 

10 

4858.3  540.3  596.7  2682.1 

4948.6  561.8  622.9  2743.3 

10 

12 

4861.3  541.0  597.5  2684.  1 

41)51.6  562.5  623.7  2745.3 

12 

14 

4864.3  541.  V  598.4  2686.1 

4954.6  563.3  624.6  2747.4 

14 

16 

4867.3  542.4  599.3  2688.2 

4957.6  584.0  625.5  2749.4 

16 

18 

4870.3  543.1  600.1  2690.2 

4960.6  564.7  626.4  2751.5 

18 

20 

4873.3  543.9  601.0  2692.3 

49G3.6  565.4  627.3  2753.5 

20 

22 

4876.3  544.6  601.9  2694.3 

4966.6  566.2  628.2  2755.6 

22 

24 

4879.4  545.3  602.7  2696.3 

4969.6  566.9  629.9  2757.7 

24 

26 

4882.4  546.0  603.6  2698.4 

4972.6  567.6  630.0  2759.7 

26 

28 

4885.4  546.7  604.5  2700.4 

4975.6  568.3  630.9  2761.8 

28 

30 

4888.4  547.4  605.3  2702.4 

4978.6  569.1  631.8  2763.8 

30 

32 

4891.4  548.1  606.2  2704.5 

4981.6  569.8  632.7  2765.9 

32 

34 

4894.4  548.8  607.0  2706.5 

4984.6  570.5  633.6  2767.9 

34 

36 

4897.4  549.5  607.9  2708.6 

4987.7  571.2  634.5  2770.0 

36 

38 

4900.4  550.2  608.8  2710.6 

4990.7  572.0  635.3  2772.0 

38 

40 

4903.5  551.0  609.7  2712.6 

4993.7  572.7  636.2  2774.1 

40 

42 

4906.5  551.7  610.5  2714.7 

4996.7  573.4  637.1  2776.2 

42 

44 

4909.5  552.4  611.4  2716.7 

4999.7  574.1  638.0  2778.2 

44 

46 

4912.5  553.1  612.3  2718.8 

5002.7  574.9  638.9  2780.3 

46 

48 

4915.5  553.8  613.2  2720.8 

5005.7  575.6  639.8  2782.3 

48 

50 

4918.5  554.5  614.1  2722.8 

5008.7  5T6.3  640.7  2784.4 

50 

52 

4921.5  555.2  614.9  2724.9 

5011.7  577.0  611.6  2786.4 

52 

54 

4924.6  555.9  615.8  2726.9 

5014.7  577.8  642.5  2788.5 

54 

56 

4927.6  556.6  616.7  2729.0 

5017.7  578.5  643.4  2790.6 

56 

58 

4930.6  557.4  617.6  2731.0 

5020.7  579.2  644.3  2792.6 

58 

60 

4933.6  558.2  618.5  2733.0 

5023.7  579.9  645.2  2794.7 

60 

52° 

53° 

L.  C.   M.   E.     T. 

L.  C.   M.   E.     T. 

0 

5023.7  579.9  645.2  2794.7 

5113.5  602.0  672.7  2856.9 

0 

2 

5026.7  580.6  6J6.1  2796.8 

5116.5  602.8  673.7  2858.9 

2 

4 

5029.7  581.3  647.0  2798.8 

5119.4  603.5  674.6  2861.0 

4 

6 

5032.7  582.1  647.9  2800.9 

5122.4  604.3  675.5  2863.1 

6 

8 

5035.7  582.8  648.9  2803.0 

5125.4  605.0  676.4  2865.2 

8 

10 

5038.7  583.5  649.8  2805.0 

5128.4  605.8  677.4  2867.3 

10 

12 

5041.7  584.3  650.7  2807.1 

5131.3  606.5  678.3  2869.4 

12 

14 

5044.7  585.0  651.6  2809.2 

5134.3  607.3  6r9.2  2871.5 

14 

16 

5047.7  585.7  652.5  2811.2 

5137.3  608.0  680.2  2873.5 

16 

18 

5050.7  586.5  653.4  2813.3 

5140.3  608.8  681.1  2875.6 

18 

20 

5053.6  587.2  654.3  2815.4 

5143.2  609.5  682.0  2877.7 

20 

23 

5056.6  587.9  655.2  2817.4 

5146.2  610.3  683.0  2879.8 

22 

24 

5059.6  588.7  656.2  2819.5 

5149.2  611.0  683.9  2881.9 

24 

26 

5062.6  589.4  657.1  2821.6 

5152.1  611.8  684.9  2884.0 

26 

28 

5065.6  590.1  658.0  2823.6 

5155.1  612.5  685.8  2886.1 

28 

30 

5068.6  590.9  658.9  2825.7 

5158.1  613.3  686.7  2888.1 

30 

32 

5071.6  591.6  659.8  2827.8 

5161.1  614.0  687.7  2890.2 

32 

34 

5074.6  592.3  660.7  2829.8 

5164.0  614.8  688.6  2892.3 

34 

36 

5077.6  593.1  661,6  2831.9 

5167.0  615.5  689.6  2894.4 

36 

'  38 

5080.6  593.8  662.5  2834.0 

5170.0  616.3  690.5  2896.5 

38 

40 

5083.6  594.5  663.5  2836.1 

5173.0  617.0  691.5  2898.6 

40 

42 

5086.6  595.3  664.4  2838.2 

5175.9  617.8  692.4  2900.7 

42 

44 

5089.6  596.0  665.3  2840.2 

5178.9  613.5  693.4  2902.8 

44 

46 

5092.6  596.7  666.2  2842.3 

5181.9  619.3  694.3  2904.9 

46 

48 

5095.6  597.5  667.2  2844.4 

5184.9  620.1  695.3  2907.0 

48 

50 

5098.6  598.2  668.1  2846.5 

5187.8  620.8  696.2  2909.1 

50 

52 

5101.6  598.9  669.0  2848.5 

5190.8  621.5  697.1  2911.2 

52 

54 

5104.6  599.7  669.9  2850.6 

5193.8  622.3  698.1  2913.3 

54 

56 

5107.6  600.4  670.9  2852.7 

5196.7  623.0  699.0  2915.4 

56 

58 

5110.6  601.2  671.8  2854.8 

5199.7  623.8  700.0  2917.5 

58 

60 

5113.5  602.0  672.7  2856.9 

5202.7  624.6  700.9  2919.5 

60 

6.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.     145 


64° 

65° 

L.  C.   M.    E.    T. 

L.  C,   M.    E.    T. 

0 

5202.7  624.6  700.9  2919.5 

5291.7  647.4  729.9  2982.8 

0 

2 

5205.7  625.4  701.9  2921.6 

5294.6  648.1  730.9  2984.9 

2 

4 

5208.6  626.1  702.8  2923.8 

5297.6  648.9  731.9  2987.1 

4 

6 

5211.6  626.9  703.8  2925.9 

5300.5  649.6  732.9  2989.2 

6 

8 

5214.6  627.6  704.8  2928.0 

5303.5  650.4  733.8  2991.3 

8 

10 

5217.5  628.4  705.7  2930.1 

5306.4  651.2  734.8  2993.4 

10 

12 

5220.5  629.2  706.7  2932.2 

5309.4  652.0  735.8  2995.5 

12 

14 

5223.5  629.9  707.7  2934.3 

5312.3  652.7  736.8  2997.7 

14 

16 

5226.4  630.7  708.6  2936.4 

5315.3  653.5  737.8  2999.8 

16 

18 

5229.4  631.4  709.6  2938.5 

5318.2  654.3  738.  7  3001.9 

18 

20 

5232.4  632.2  710.5  2940.6 

5321.2  655.1  739.7  3004.0 

20 

22 

5235.3  633.0  711.5  2942.7 

5324.1  655.8  740.7  3006.2 

22 

24 

5238.3  633.7  712.5  2944.8 

5327.1  656.6  741.7  3008.3 

24 

26 

5241.3  634.5  713.4  2946.9 

5330.0  657.4  742.7  3010.4 

26 

28 

5244.2  635.2  714.4  2949.0 

5333.0  658.2  743.7  3012.5 

28 

30 

5247.2  636.0  715.3  2951.1 

5335.9  658.9  744.7  3014.7 

30 

32 

5250.2  636.8  716.3  2953.2 

5338.8  659.7  745.7  3016.8 

32 

34 

5253.1  637.5  717.3  2955.3 

5341.8  660.5  746.7  3018.9 

34 

36 

5256.1  638.3  718.2  2957.5 

5344.7  661.3  747.7  3021.1 

36 

38 

5259.1  639.0  719.2  2959.6 

5347.7  662.0  748.7  3023.2 

38 

40 

5262.0  639.8  720.2  2961.7 

5350.6  662.8  749.7  3025.3 

40 

42 

5265.0  640.6  721.1  2963.8 

5353.6  663  6  750.7  3027.5 

42 

44 

5268.0  641.3  722.1  2965.9 

5356.5  664.4  751.7  3029.6 

44 

46 

5270.9  642.1  723.1  2968.0 

5359.5  665.1  752.6  3031.7 

46 

48 

5273.9  642.8  724.1  2970.1 

5362.4  665.9  753.6  3033.8 

48 

50 

5276.9  643.6  725.0  2972.2 

5365.4  666,7  754.6  3036.0 

50 

52 

5279.8  644.4  726.0  2974.4 

5368.3  667.5  755.6  3038.1 

52 

54 

5282.8  645.1  727.0  2976.5 

5371.3  668.3  756.6  3040.2 

54 

56 

5285.8  645.9  728.0  2978.6 

5374.2  669.1  757.6  3042.4 

56 

58 

5288.7  646.6  729.0  2980.7 

5377.2  669.9  758.6  3044.5 

58 

60 

5291.7  647.4  729  9  2932.8 

5380.1  670.7  759.6  3046.6 

60 

66° 

57° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

/ 

0 

5380.1  670.7  759.6  3016.6 

5468.2  694.4  790.2  3111.1 

0 

2 

5383.0  671.4  760.6  3048.8 

5471.1  695.2  791.2  3113.3 

2 

4 

5386.0  672.2  761.6  3050.9 

5474.0  696.0  792.2  3115.4 

4 

6 

5388.9  672.9  762.7  3053.1 

5477.0  696.8  793.3  3117.6 

6 

8 

5391.8  673.7  763.7  3055.2 

5479.9  697.6  794.3  3119.7 

8 

10 

5394.8  674.4  764.7  3057.4 

5482.8  698.4  795.3  3121.9 

10 

12 

5397.7  675.2  765.7  3059.5 

5485.7  699.2  796.3  3124.1 

12 

14 

5400.7  676.0  766.7  3061.6 

5488.7  700.0  797.4  3126.2 

14 

16 

5403.6  676.8  767.7  3063.8 

5491.6  700.8  798.4  3128.4 

16 

18 

5406.5  677.6  768.7  3065.9 

5494.5  701.6  799.4  3130.6 

18 

20 

5409.5  678.4  769.7  3068.1 

5497.4  702.4  800.5  3132.7 

20 

22 

5412.4  679.2  770.8  3070.2 

5500.3  703.2  801.5  3134.9 

22 

24 

5415.3  680.0  771.8  30^2.4 

5503.3  704.0  802.6  3137.0 

24 

26 

5418.3  680.8  772.8  3074.5 

5506.2  704.8  803.6  3139.2 

26 

28 

5421.2  681.6  773.8  3076.6 

5509.1  705.6  804.7  3141.4 

28 

30 

5424.1  682.4  774.8  3078.8 

5512.0  706.4  805.7  3143.5 

30 

32 

5427.1  683.2  775.8  3080.9 

5515.0  707.2  806.8  3145.7 

32 

34 

5430.0  684.0  776.8  3083.1 

5517.9  708.0  807.8  3147.9 

34 

36 

5433.0  684.8  777.8  3085.2 

5520.8  708.8  808.8  3150.0 

36 

38 

5435.9  685.6  778.9  3087.4 

5523.7  709.6  809.9  3152.2 

38 

40 

5438.8  686.4  779.9  3089.6 

55-26.7  710.4  810.9  3154.4 

40 

42 

5441.8  687.2  780.9  3091.7 

5529.6  711.2  812.0  3156.6 

42 

44 

5444.7  688.0  781.9  3093.9 

5532.5  712.0  813.0  3158.7 

44 

46 

5447.6  688.8  783.0  3096.0 

5535.4  712.8  814.1  3160.9 

46 

48 

5450.6  689.6  784.0  3098.2 

5538.4  713.6  815.1  3163.1 

48 

50 

5453.5  690.4  785.0  3100.3 

5541.3  714.4  816.2  3165.3 

50 

52 

5456.5  691.2  786.0  3102.5 

5544.2  715.2  817.2  3167.4 

52 

54 

5459.4  692.0  787.1  3104.6 

5547.1  716.0  818.3  3169.6 

54 

56 

5462.3  692.8  788.1  3106.8 

5550.0  716.8  819.3  3171.8 

56 

58 

5465.3  693.6  789.1  3108.9 

5553.0  717.6  820.4  3174.0 

58 

60 

5468.2  694.4  790.2  3111.1 

5555.9  718.4  821.4  3176.1 

60 

146        6.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


68° 

69° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

5555.9  718.4  821.4  3176.1 

5643.1  742.8  853.5  3241.9 

0 

2 

5558.8  719.2  822.5  3178.3 

5646.0  743.6  854.6  3244.1 

2 

4 

5561.7  720.0  823.5  3180.5 

5648.9  744.4  855.7  3246.3 

4 

6 

5564.6  720.8  824.6  3182.7 

5651.8  745.3  856.8  3248.5 

6 

8 

5567.5  721.6  825.7  3184.9 

5654.7  746.1  857.9  3250.7 

8 

10 

5570.4  7224  826.7  3187.1 

5657.6  746.9  859.0  3252.9 

10 

12 

5573.3  723.2  827.8  3189.2 

5660.5  747.7  860.0  3255.1 

12 

14 

5576.2  724.0  828.9  3191.4 

5663.4  748.6  861.1  3257.3 

14 

16 

5579.2  724.8  829.9  3193.6 

5666.3  749.4  862.2  3259.5 

16 

18 

5582.1  725.6  831.0  3195.8 

5669.2  750.2  863.3  3261.7 

18 

20 

5585.0  726.5  832.1  3198.0 

5672.1  751.1  864.4  3263.9 

20 

22 

5587.9  727.3  833.1  3200.2 

5675.0  751.9  865.5  3266.1 

22 

24 

5590.8  728.1  834.2  3202.4 

5677.9  752.7  866.6  3268  3 

24 

26 

5593.7  728.9  835.3  3204.5 

5680.8  753.5  867.7  3270.5 

26 

28 

5596.6  729.7  836.3  3206.7 

5683.7  754.4  868.8  3272.7 

28 

30 

5599.5  730  5  837.4  3208.9 

5686.5  755.2  869.9  3274.9 

30 

32 

5602.4  731.3  838.4  3211.1 

5689.4  756.0  871.0  3277.1 

32 

34 

5605.3  732.1  839.5  3213.3 

5692.3  756.9  872.1  3279.4 

34 

36 

5608.2  732.9  840.6  3215.5 

5695.2  757.7  873.2  3281  6 

36 

38 

5611.1  733.7  841.6  3217.7 

5698.1  758.5  874.3  3283.8 

38 

40 

5614.0  734.6  842  7  3219.9 

5701.0  759.4  875.4  3286.0 

40 

42 

5616.9  735.4  843.8  3222.1 

5703.9  760.2  876.5  3288.2 

42 

44 

5619.8  736.2  844.9  3224.3 

5706.8  761.0  877.6  3290.5 

44 

46 

5622.8  737.0  846.0  3226.5 

57097  761.9  878.7  3292.7 

46 

48 

5625.7  737.8  847.0  3228.7 

5712  6  762.7  879.8  3294.9 

48 

50 

5628.6  738.6  848.1  3230.9 

5715.5  763.5  880.9  3297.1 

50 

52 

5631.5  739.4  849.2  3233.1 

5718.4  764.4  882.0  3299.3 

52 

54 

5634.4  740.2  850.3  3235.3 

5721.3  765.2  883.1  3301.5 

54 

56 

5637.3  741.0  851.4  3237.5 

5724.2  766.0  884.2  8303.8 

56 

58 

5640.2  741.9  852.5  32-39.7 

5727.1  766.8  885.3  3306.0 

58 

60 

5643.1  742.8  853.5  3241.9 

5730.0  767.7  886.4  3308.2 

60 

60° 

61° 

L.  C.   M,    E.    T. 

L.  C.   M.    E.    T. 

0 

5730.0  767.7  886.4  3303.2 

5816.4  792.9  920.2  3375.2 

0 

2 

5732.9  768.5  887.5  3310.4 

5*19.3  793.7  921.4  3377.4 

2 

4 

5735.8  769.4  888.7  3312.7 

5822.1  794.6  922.5  3379.7 

4 

6 

5738.6  770.2  889.8  3314.9 

5825.0  795.4  923.6  3381.9 

6 

8 

5741.5  771.1  890.9  3317.1 

5827.9  796.3  924.8  3384,2 

8 

10 

5744.4  771.9  892.0  3319.3 

5830.7  797.1  9:25.9  3386.  4 

10 

13 

5747.3  772.7  893.1  3321.6 

5833.6  798.0  927.1  3388.7 

12 

14 

5750.2  773.6  894.3  3323.8 

5836.5  798.8  928.2  3390.9 

14 

16 

5753.0  774.4  895.4  3326.0 

5839.3  799.7  9'29.3  3393.2 

16 

18 

5755.9  775.3  896.5  3328.3 

5842.2  800.5  930.5  3395.4 

18 

20 

5758.8  776.1  897.6  3330.5 

5845.1  801.4  931.6  3397.7 

20 

22 

5761.7  776.9  898.8  3332.7 

5847.9  802.2  932.8  8399.9 

22 

24 

5764.6  777.8  899.9  3334.9 

5850.8  803.1  933.9  3402.2 

24 

26 

5767.4  778.6  901.0  3337.2 

5853.7  803.9  935.1  3404.4 

26 

28 

5770.3  779.5  902.1  3339.4 

5856.5  804.8  936.3  3406.7 

28 

30 

5773.2  780.8  903.2  3341.6 

5859.4  805.6  937.4  3408.9 

30 

32 

5776.1  781.1  904.4  3343.9 

5862.3  806.5  938.6  3411.2 

32 

34 

5779.0  782.0  905.5  3346.1 

5865.1  807.3  939.7  3413.5 

34 

36 

5781.8  782.8  906.6  3348.3 

5868.0  808.2  940.9  3415.7 

36 

38 

5784.7  783.7  907.7  3350.6 

5870.9  809.0  942.1  3418.0 

38 

40 

5787.6  784.5  908.8  3352.8 

5873.7  809.9  943.2  3420.3 

40 

42 

5790.5  785.3  910.0  3355.0 

5876.6  810.7  944.4  3422.5 

42 

44 

5793.4  786.2  911.1  3357.3 

5879.5  811.6  945.5  3424.8 

44 

46 

5796.2  787.0  912.3  3359.5 

5882.3  812.4  946.7  3427.1 

46 

48 

5799.1  787.9  913.4  3361.8 

5885.2  813.3  947.8  3429.3 

48 

50 

5802.0  788.7  914.5  3364.0 

5888.1  814.1  949.0  3431.6 

50 

52 

5804.9  789.5  915.7  3366.2 

5890.9  815.0  950.2  3433.9 

52 

54 

5807.8  790.4  916.8  3368.5 

5893.8  815.8  951.3  3436.1 

54 

56 

5810.6  791.2  918.0  3370.7 

5896.7  816.7  952.5  3438.4 

56 

58 

5813.5  792.1  919.1  3373.0 

5899.5  817.5  953.6  3440.7 

58 

L  60 

5816.4  792.9  920.2  3375.2 

5902.4  818.4  954.8  3442.9 

60  , 

6.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.     147 


62° 

63° 

/ 

L.  C.   M.    E.    T. 

L.  C.    M.    E.    T. 

0 

5902.4  818.4  954.8  3442.9 

5987.8  844.4   990.3  3511.3 

0 

2 

5905.2  819.3  956.0  3445.2 

5990.6  845.3   991.5  3513.6 

2 

4 

5908.1  820.1  957.2  3447.5 

5993.5  846.2   992.7  3515.9 

4 

6 

5910.9  821.0  9583  3449.7 

5996.3  847.1   993.9  3518.2 

6 

8 

5913.8  821.8  959.5  3452.0 

5999.1  847.9   995.1  3520.5 

8 

10 

5916.6  822.7  960.7  3454.3 

6002.0  848.8   996.3  3522.8 

10 

12 

5919.5  823.6  961.9  3456.6 

6004.8  849.7   997.5  3525.1 

12 

14 

5922.3  824.4  963.0  3458.8 

6007.7  850.6   998.7  3527.4 

14 

16 

5925.2  825.3  964.2  3461.1 

6010.5  851.4   999.9  3529.7 

16 

is 

5928.0  826.1  965.4  3463.4 

6013.3  852.3  1001.1  3532.0 

18 

20 

5930.9  827.0  966.6  3465.7 

6016.2  853.2  1002.3  3534.3 

20 

22 

5933.7  827.9  967.8  3167.9 

6019.0  854.1  1003.5  3536.6 

22 

24 

5936.6  828.7  968.9  3470.2 

6021.8  854.9  1004.7  3538.9 

24 

26 

5939.4  829.6  970.1  3472.5 

6024.7  855.8  1005.9  3541.2 

26 

28 

5942.3  830.4  971.3  3474.7 

6027.5  856.7  1007.1  3543.5 

28 

30 

5945.1  831.3  972.5  3477.0 

6030.3  857.6  1008.4  3545.8 

30 

82 

5947.9  832.2  973.6  3479.3 

6033.2  858.4  1009.6  3548.1 

32 

34 

5950.8  833.0  974.8  3481.6 

6036.  0  859.3  1010.8  3550.4 

34 

36 

5953.6  833.9  976.0  3483.9 

6038.9  860.2  1012.0  3552.7 

36 

38 

5956.5  834.7  977.2  3486.2 

6041.7  861.1  1013.2  3555.0 

38 

40 

5959.3  835.6  978.4  3488.5 

6044.5  861.9  1014.5  3557.3 

40 

42 

5962.2  836.5  979.6  3490.7 

6047.4  862.8  1015.7  3559.6 

42 

44 

5965.0  837.4  980.8  3493.0 

6050.2  863.7  1016.9  3562.0 

44 

46 

5967.9  838.3  982.0  3495.3 

6053.0  864.6  1018.1  3564.3 

46 

48 

5970.7  839.1  983.2  3497.6 

0055.9  865.4  1019.3  3566.6 

48 

50 

5973.6  840.0  984.4  3499.9 

60f>8.7  866.3  1020.6  3568.9 

50 

52 

5976.4  840.9  985.5  3502.2 

6061.6  867.2  1021.8  3571.2 

52 

54 

5979.3  84f.7  986.7  3504.5 

60(54.4  868.1  1023.0  3573.5 

54 

56 

5982.1  842.6  9S7.9  3506.8 

6067.2  868.9  1024.2  3575.8 

56 

58 

5985.0  843.5  989.1  3509.0 

6070.1  869.8  1025.4  3578.1 

58 

60 

5987.8  844.4  990.3  3511.3 

6072.9  870.7  1026.7  3580.4 

60 

64°: 

65° 

L.  C.   M.  ^   E.    T. 

L.  C.   M.    E.    T. 

0 

6072.9  870.7  1026.7  3580.4 

6157.5  897.3  1064.0  3650.4 

0 

2 

6075.7  871.5  1027.9  3582.8 

6160.3  898.2  1065.2  3652.8 

2 

4 

6078.5  872.4  1029.2  3585.1 

6163.1  899.1  1066.5  3655.1 

4 

6 

6081.4  873.3  1030.4  3587.4 

6165.9  900.0  1067.7  3657.5 

6 

8 

6084.2  874.2  1031.7  3589.7 

6168.7  900.9  1069.0  3659.8 

8 

10 

6087.0  875.1  1032.9  3592.1 

6171.5  901.8  1070.2  3662.2 

10 

12 

6089.8  875.9  1034.1  3594.4 

6174.3  902.7  1071.5  3664.5 

12 

14 

6092.6  876.8  1035.4  3596.7 

6177.1  903.6  1072.7  3666.9 

14 

16 

6095.5  877.7  1036.6  3599.1 

6179.9  904.5  1074.0  3669.2 

16 

18 

6098.3  878.6  1037.9  3601.4 

6182.7  905.4  1075.2  3671.6 

18 

20 

6101.1  879.5  1039.1  3603.7 

6185.5  906.3  1076.6  3673.9 

20 

22 

6103.9  880.3  1040.3  3606.0 

6188.3  907.2  1077.8  3676.2 

22 

24 

6106.7  881.2  1041.6  3608.4 

6191.1  908.1  1079.1  3678.6 

24 

26 

6109.6  882.1  1042.8  3610.7 

6193.9  909.0  1080.4  3680.9 

26 

28 

6112.4  883.0  1044.1  3613.0 

6196.7  909.9  1081.7  3683.3 

28 

30 

6115.2  883.9  1045.3  3615.3 

6199.5  910.8  1083.0  3685.6 

30 

32 

6118.0  884.7  1046.5  3617.7 

6202.3  911.7  1084.2  3688.0 

32 

34 

6120.8  8S5.6  1047.8  3620.0 

6205.1  912.6  1085.5  3690.4 

34 

36 

6123.7  886.5  1049.0  3622.3 

6208.0  913.5  1086.8  3692.7 

36 

38 

6126.5  887.4  1050.3  3624.7 

6210.8  914.4  1088.1  3695.1 

38 

40 

6129.3  888.3  1051.5  3627.0 

6213.6  915.3  1089.4  3697.4 

40 

42 

6132.1  889.2  1052.7  3629.4 

6216.4  916.2  1090.6  3699.8 

42 

44 

6134.9  890.1  1054.0  3631.7 

6219.2  917.1  1091.9  3702.2 

44 

46 

6137.8  891.0  1055.2  3634.0 

6222.0  918.0  1093.2  3704.5 

46 

48 

6140.6  891.9  1056.5  3636.4 

6224.8  918.9  1094.5  3706.9 

48 

50 

6143.4  892.8  1057.7  3638.7 

6227.6  919.8  1095.8  3709-3 

50 

52 

6146.2  893.7  1059.0  3641.1 

6230.4  920.7  1097.0  3711.6 

52 

54 

6149.0  894.6  1060.2  3643.4 

6233.2  921.6  1098.3  3714.0 

54 

56 

6151.9  895.5  1061.5  3645.7 

6236.0  922.5  1099.6  3716.3 

56 

58 

6154.7  896.4  1062.7  3648.1 

6238.8  923.4  1100.9  3718.7 

58 

60 

6157.5  897.3  1064.0  3650.4 

6241.6  924.3  1102.2  3721.1 

60 

'148    6. -FUNCTIONS   OF  A  ONE-DEGREE   CURVE. 


!  , 

66° 

67° 

L.  C.   M.    E.     T. 

L.  C.   M.    E.    T. 

0 

6241.6  924.3  1102.2  3721.1 

6325.2  951.8  1141.5  3792.6 

0 

2 

6244.4  925.2  1103.5  3723.4 

6328.0  952.7  1142.8  3795.0 

2 

4 

6247.2  926.1  1104.8  3725.8 

6330.7  953.6  1144.1  3797.4 

4 

6 

6250.0  927.0  1106.1  3728.2 

6333.5  954.5  1145.4  3799.8 

6 

8 

6252.7  927.9  1107.4  3730.6 

6336.3  955.5  1146.7  3802.2 

8 

10 

6255.5  928.8  1108.7  3732.9 

6339.0  956.4  1148.1  3804.6 

10 

12 

6258.3  929.8  1110.0  3735.3 

6341.8  957.3  1149.4  3807.0 

12 

14 

6261.1  930.7  1111.3  3737.7 

6344.6  958.2  1150.7  3809.4 

14 

16 

6263.9  931  6  1112.6  3740.1 

6347.4  959.2  1152.0  3811.8 

16 

18 

6266.7  932.5  1113.9  3742.4 

6350.1  960.1  1153.3  3814.2 

18 

20 

6269.5  933.4  1115.2  3744.8 

6352.9  961.0  1154.7  3816.6 

20 

22 

6272.3  934.3  1116.5  3747.2 

6355.7  961.9  1156.0  3819.0 

22 

24 

6275,0  935.3  1117.8  3749.6 

6358.4  962.9  1157.4  3821.4 

24 

26 

6277.8  936.2  1119.1  3751.9 

6301.2  963.8  1158.7  3823.8 

26 

28 

6280.6  937.1  1120.4  3754.3 

6364.0  964.7  1160.1  3826.2 

28 

30 

6283.4  938.0  1121.7  3756.7 

6366.7  965.6  1161.4  3828.6 

30 

32 

6286.2  938.9  1123.0  3759.1 

6369.5  966.6  1162.8  3831.0 

32 

34 

6289.0  939.8  1124.3  3761.5 

0372.3  907.5  1164.1  3833.4 

34 

36 

6291.8  940.8  1125.6  3763.9 

6375.1  968.4  1165.5  3835.9 

36 

38 

6294.5  941.7  1126.9  3766.3 

6377.8  909.3  1166.8  3838.3 

38 

40 

6297.3  942  6  1128.3  3768.7 

6380.6  970.3  1168.2  3840.7 

40 

42 

6300.1.  943.5  1129.6  3771.0 

6383.4  971.2  1169.5  3843.1 

42 

44 

6302.9  944.4  1130.9  3773.4 

6386.1  972.1  1170.9  3845.5 

44 

46 

6305.7  945.3  1132.2  3775.8 

6388.9  973.0  1172.2  3847.9 

46 

1.48 

6308.5  946.3  1133.5  3778.2 

6391.7  974.0  1173.6  3850.4 

48 

50 

6311.3  947.2  1134.9  3780.6 

6394.4  974.9  1174.9  3852.8 

50 

52 

6314.1  948.1  1136.2  3783.0 

6397.2  975.8  1176.3  3855.2 

52 

54 

6316.8  949.0  1137.5  3785.4 

6400.0  976.8  1177.6  3857.6 

54 

56 

6319.6  949.9  1138.8  3787.8 

6402.8  977.7  1179.0  3860.0 

56 

58 

6322.4  950.8  1140.1  3790.2 

6405.5  978.6  1180.3  3862.5 

58 

60 

6325.2  951.8  1141.5  3792.6 

6408.3  979.6  1181.6  3864.9 

60 

68° 

69° 

L.  C.   M.    E.    T. 

L.  C.    M.  *  E.    T. 

0 

6408.3  979.6  1181.6  3864.9 

6491.1  1007.7  1222.9  3938.1 

0 

2 

6411.1  980.5  1183.0  3807.3 

6493.8  1008.7  1224.3  3940.6 

2 

4 

6413.8  981.4  1184.4  3869.7 

6496.6  1009.6  1225.7  3943.0 

4 

6 

6416.  6  982.4  1185.7  3372.2 

6499.3  1010.6  1227.1  3945.5 

6 

8 

G419.3  983.3  1187.1  3874.6 

6502.1  1011.5  1228.5  3947.9 

8 

10 

61-22.1  984.2  1188.5  3877.0 

6504.8  1012.5  1229.9  3950.4 

10. 

12 

0421.  9  985.2  1189.8  3879.5 

6507.5  1013.4  1231.3  3952.9 

12 

14 

6427.6  986.1  1191.2  3881.9 

6510.3  1014.4  1232.7  3955.3 

14 

10 

6430.4  987.0  1192.6  3884.3 

6513.0  1015.3  1234.1  3957.8 

16 

18 

6433.1  988.0  1193.9  3886.8 

6515.8  1016.3  1235.5  3960.2 

18 

20 

6435.9  988.9  1195.3  3889.2 

6518.5  1017.2  1236.9  3962.7 

20 

22 

6438.7  980.8  1196.7  3891.6 

6521.2  1018.2  1238.3  3965.2 

22 

24 

6141.4  990.8  1198.0  3894.1 

6524.0  1019.1  1239.7  3967.6 

24 

26 

6444.2  991.7  1199.4  3896.5 

0520.7  1020.1  1241.1  3970,1 

26 

28 

6140.9  992.6  1200.8  3898.9 

6529.5  1021.0  1242.5  3972.5 

28 

30 

6149.7  99J.  6  1202.1  3901.4 

0532.2  1022.0  1243.9  3975.0 

30 

32 

6452.5  994.5  1203.5  3903.8 

0534.9  10.  '2.  9  1245.3  3977.5 

32 

34 

6455.2  995.4  1204.9  3906.3 

0537.7  1023.9  1246.7  3980.0 

34 

36 

0458.0  996.4  1206.2  3908.7 

0540.4  1024.8  1248.1  3982.4 

36 

38 

6400.7  997.3  1207.6  3911.2 

6543.2  1025.8  1249.5  3984.9 

38 

40 

6463.5  908.2  1209.0  3913.6 

0545.9  1026.7  1250.9  3987.4 

40 

42 

0466.3  999.2  1210.3  3916.1 

6548.6  1027.7  1252.3  3989.9 

42 

44 

6469.0  1000.1  1211.7  3918.5 

6551.4  1028.6  1253.7  3993.3 

44 

46 

6471.8  1001.0  1213.1  3921.0 

CS54.1  1029.6  1255.1  3994.8 

46 

48 

6474.5  1002.0  1214.5  3923.4 

6550.9  1030.5  1256.5  3997.3 

48 

50 

6477.3  1002.9  1215.9  3925.9 

6559.6  1031.5  1257.9  3999.8 

50 

52 

6480.1  1003.8  1217.3  3928.3 

6562.3  1032.4  1259.3  4002.2 

52 

54 

6482.8  1004.8  1218.7  3930.8 

6565.1  1033.4  1200.7  4004.7 

54 

56 

6485.6  1005.7  1220.1  3933.2 

6567.8  1034.3  1262.1  4007.2 

56 

58 

6488.3  1006.7  1221.5  3935.7 

6570.6  1035.3  1263.5  4009.7 

58 

60 

6491.1  1007.7  1222.9  3938.1 

6573.3  1036.3  1265.0  4012.1 

60 

6.— FUNCTIONS  OF  A  ONE-DEGREE   CURVE.    149 


70° 

71° 

L.  C.    M.    E.     T. 

L.  C.    M.    E.    T. 

t 

0 

6573.3  1036.3  1265.0  4012.1 

6654.9  1065.1  1308.4  4087.1 

0 

2 

6576.0  1037.3  1266.4  4014.6 

6657.6  1066.1  1309.9  4089.7 

2 

4 

6578.7  1038.2  1267.9  4017.1 

6660.3  1067.0  1311.3  4092.2 

4 

6 

6581.5  1039.2  1269.3  4019.6 

6663.0  1068.0  1312.8  4094.7 

6 

8 

6584.2  1040.1  1270.8  4022.1 

6605.7  1068.9  1314.2  4097.2 

8 

10 

6586.9  1041.1  1272.2  4024.6 

6668.4  1069.9  1315.7  4099.8 

10 

12 

6589.6  1042.1  1273.6  4027.1 

6671.1  1070.9  1317.2  4102.3 

12 

14 

6592.3  1043.0  1275.1  4029.6 

6673.8  1071.9  1318.6  4104.8 

14 

16 

6595.1  1044.0  1276.5  4032.1 

6676.6  1072.9  1320.1  4107.3 

16 

18 

6597.8  1044.9  1278.0  4034.6 

6679.3  1073.8  1321.5  4109.8 

18 

20 

6600.5  1045.9  1279.4  4037.1 

6682.0  1074.8  1323.0  4112.4 

20 

22 

6603.2  1046.9  1280.8  4039.6 

6684.7  1075.8  1324.4  4114.9 

22 

24 

6605.9  1047.8  1282.3  4042.1 

6687.4  1076.8  1325.9  4117.4 

24 

26 

6608.7  1048.8  1283.7  4044.6 

6690.1  1077.7  1327.4  4119.9 

26 

28 

6611.4  1049.7  1285.2  4047.1 

6692.8  1078.7  1328.9  4122.4 

28 

30 

6614.1  1050.7  1286.6  4049.6 

6695.5  1079.7  1330.4  4125.0 

30 

32 

6616.8  1051.7  1288.0  4052.1 

6098.2  1080.7  1331.8  4127.5 

32 

34 

6619.5  1052.6  1289.5  4054.6 

6700.9  1081.6  1333.3  4130.1 

34 

36 

6622.3  1053.6  1290.9  4057.1 

6703.6  1082.6  1334.8  4132.6 

36 

38 

6625.0  1054.5  1292.4  4059.6 

6706.3  1083.6  1336.3  4135.1 

38 

40 

6627.7  1055.5  1293.8  4062.1 

6709.0  1084.5  1337.8  4137.7 

40 

42 

6630.4  1056.5  1295.3  4004.6 

6711.7  1085.5  1339.2  4140.2 

42 

44 

6633.1  1057.4  1296.7  4067.1 

6714.4  1086.5  1340.7  4142.7 

44 

46 

6635.9  1058.4  1298.2  4069.6 

6717.2  1087.5  1342.2  4145.3 

46 

48 

6638.6  1059.3  1299.6  4072.1 

6719.9  1088.4  1343.7  4147.8 

48 

50 

6641.3  1060.*  1301.1  4074.6 

6722.6  1089.4  1345.2  4150.4 

50 

52 

6644.  C  1061.3  1302.6  4077.1 

6725.3  1090.4  1346.7  4152.9 

52 

54 

6646.7  1062.2  1304.0  4079.6 

6728.0  1091.3  1348.2  4155.4 

54 

56 

6649.5  1063.2  1305.5  4082.1 

6730.7  1092.3  1349.7  4158.0 

56 

58 

6652.2  1064.1  1306.9  4084.6 

6733.4  1093.3  1351.2  4160.5 

58 

60 

6654.9  1065.1  1308.4  4087.1 

6736.1  1094.3  1352.7  4163.1 

60 

1  1.° 

73° 

L.  C.    M.     E.    T. 

L.  C.    M,    E.    T. 

0 

6736.1  1094.3  1352.7  4163.1 

6816.6  1123.9  1398.1  4240.0 

0 

2 

6738.8  1095.2  1354.2  4165.6 

6819.3  1124.8  1399.6  4242.6 

2 

4 

6741.5  1096.2  1355.7  4168.2 

6821.9  1125.8  1401.2  4245.1 

4 

6 

6744.1  1097.2  1357.2  4170.7 

6824.6  1126.8  1402.7  4247.7 

6 

8 

6746.8  1098.2  1358.7  4173.3 

6827.3  1127.8  1404.2  4250.3 

8 

10 

6749.5  1099.2  1360.2  4175.8 

6830.0  1128.8  1405.8  4252.9 

10 

12 

6752.2  1100.1  1361.7  4178.4 

6832.6  1129.8  1407.3  4255.5 

12 

14 

6754.9  1101.1  1303.2  4181.0 

6835.3  1130.8  1408.8  4258.1 

14 

16 

6757.6  1102.1  1364.7  4183.5 

6838.0  1131.8  1410.4  4260.7 

16 

18 

6760.2  1103.1  1366.2  '4186.1 

6840.7  1132.8  1411.9  4263.2 

18 

20 

6762.9  1104.1  1367.7  4188.6 

6843.3  1133.8  1413.5  4265.8 

20 

22 

6765.6  1105.1  1369.2  4191.2 

6846.0  1134.8  1415.1  4268.4 

22 

24 

6768.3  1106.0  1370.7  4193.7 

6848.7  1135.8  1416.6  4271.0 

24 

26 

6771.0  1107.0  1372.2  4196.3 

6851.3  1136.8  1418.2  4273.6 

26 

28 

6773.7  1108.0  1373.7  4198.8 

6854.0  1137.8  1419.7  4276.2 

28 

30 

6776.3  1109.0  1375.2  4201.4 

6856.7  1138.8  1421.3  4278.8 

30 

32 

6779.0  1109.9  1376.7  4204.0 

6859.4  1139.8  1422.9  4281.4 

32 

34 

6781.7  1110.9  1378.2  4206.5 

6862.0  1140.8  1424.4  4284.0 

34 

36 

6784.4  1111.9  1379.7  4209.1 

6804.7  1141.8  1426.0  4286.6 

30 

38 

6787.1  1112.9  1381.2  4211.7 

6867.4  1142.8  1427.5  4289.2 

38 

40 

6789.8  1113.9  3382.8  4214.3 

6870.1  1143.8  1429.1  4291.8 

40 

42 

6792.4  1114.9  1384.3  4216.8 

6872.7  1144.8  1430.7  4294.4 

42 

44 

0795.1  1115.9  1385.8  4219.4 

6875.4  1145.8  1432.2  4297.0 

44 

46 

6797.8  1116.9  1387.4  4222.0 

6878.1  1146.8  1433.8  4299.6 

46 

48 

6800.5  1117.9  1388.9  4224.5 

6880.8  1147.8  1435.3  4302.2 

48 

50 

6803.2  1118.9  1390.4  4227.1 

6883.4  1148  8  1436.9  4304.8 

50 

52 

6805.9  1119.9  1392.0  4229.7 

6886.1  1149.8  1438.5  4307.4 

52 

54 

6808.5  1120.9  1393.5  4232.3 

68S8.8  1150.8  1440.0  4310.0 

54 

56 

6811.2  1121.9  1395.0  4234.8 

6891.4  1151.8  1441.6  4312.6 

56 

58 

6813.9  1122.9  1396.6  4237.4 

6894.1  1152.8  1443.1  4315.2 

58 

80 

6816.6  1123.9  1398.1  4240.0 

6896.8  1153.8  1444.7  4317.8 

60 

150       6.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


74° 

75° 

L.  C.    M.    E.     T. 

L.  C.    M.    E.     T. 

0 

6896.8  1153.8  1444.7  4317.8 

6976.4  1184.1  1492.5  4396.7 

0 

2 

6899.4  1154.8  1446.2  4320.5 

6979.0  1185.1  1494.1  4399.4 

2 

4 

6902.1  1155.8  1447.8  4323.1 

6981.7  1186.1  1495.7  4402.1 

4 

6 

6904.8  1156.8  1449.4  4325.7 

6984.3  1187.1  1497.3  4404.7 

6 

8 

6907.4  1157.8  1451.0  4328.3 

6986.9  1188.1  1499.0  4407.4 

8 

10 

6910.1  1158.8  1452.6  4330.9 

6989.6  1189.2  1500.6  4410.0 

10 

18 

6912.7  1159.8  1454.1  4333.6 

6992.2  1190.2  1502.2  4412.7 

12 

14 

6915.4  1160.8  1455.7  4336.2 

6994.9  1191.2  1503.8  4415.3 

14 

16 

6918.0  1161.8  1457.3  4338.8 

6997.5  1192.2  1505.4  4418.0 

16 

18 

6920.7  1162.8  1458.9  4341.4 

7000.1  1193.2  1507.0  4420.7 

18 

20 

6923.3  1163.9  1460.5  4344.0 

7002.8  1194.3  1508.7  4423.3 

20 

22 

6926.0  1164.9  1462.0  4346.7 

7005.4  1195.3  1510.3  4426.0 

22 

24 

6928.6  1165.9  1463.6  4349.3 

7008.0  1196.3  1512.0  4428.6 

24 

26 

6931.3  1166.9  1465!2  4351.9 

7010.7  1197.3  1513.6  4431.3 

26 

28 

6933.9  1167.9  1466.8  4354.5 

7013.3  1198.3  1515.3  4434.0 

28 

30 

6936.6  1168.9  1468.4  4357.1 

7015.9  1199.4  1516.9  4436.6 

30 

32 

6939.2  1169.9  1469.9  4359.8 

7018.6  1200.4  1518.5  4439.3 

32 

34 

6941.9  1170.9  1471.5  4362.4 

7021.2  1201.4  1520.2  4442.0 

34 

36 

6944.6  1171.9  1473.1  4365.1 

7023.9  1202.4  1521.8  4444.6 

36 

88 

6947.2  1172.9  1474.7  4367.7 

7026.5  1203.4  1523.5  4447.3 

38 

40 

6949.9  1174.0  1476.4  4370.3 

7029.1  1204.5  1525.1  4450.0 

40 

42 

6952.5  1175.0  1478.0  4373.0 

7031.8  1205.5  1526.7  4452.7 

42 

44 

6955.2  1176.0  1479.6  4375.6 

7034.4  1206.5  1528.4  4455.3 

44 

46 

6957.8  1177.0  1481.2  4378.3 

7037.0  1207.5  1530.0  4458.0 

46 

48 

6960.5  1178.0  1482.8  4380.9 

7039.7  1208.5  1531.7  4460.7 

48 

50 

6963.1  1179.0  1484.4  4383.5 

7042.3  1209.6  1533.3  4463.4 

50 

52 

6965.8  1180.0  1486.0  4386.2 

7045.0  1210.6  1534.9  4466.0 

52 

54 

6968.4  1181.0  1487.7  4388.8 

7047.6  1211.6  1536.6  4468.7 

54 

56 

6971.1  1182.0  1489.3  4391.5 

7050.2  1212.6  1538.2  4471.4 

56 

58 

6973.7  1183.0  1490.9  4394.1 

7052.9  1213.6  1539.9  4474.1 

58 

60 

6976.4  1184.1  1492.5  4396.7 

7055.5  1214.7  1541.5  4476.7 

60 

76° 

77° 

L.  C.    M.    E.     T. 

L.  C.    M    E.     T. 

0 

7055.5  1214.7  1541.5  4476,7 

7134.0  1245.6  1591.7  4557.8 

0 

2 

7058.1  1215.7  1543.2  4479.4 

7136.6  1246.6  1593.4  4560.5 

2 

4 

7060.7  1216.7  1544.9  4482.1 

7139.2  1247.7  1595.1  4563.3 

4 

6 

7063.3  1217.8  1546.5  4484.8 

7141.8  1248.7  1596.8  4566.0 

6 

8 

7066.0  1218.8  1548.2  4487.5 

7144.4  1249.8  1598.5  4568.7 

8 

10 

7068.6  1219.8  1549.9  4490.2 

7147.0  1250.8  1600.2  4571.5 

10 

12 

7071.2  1220.9  1551.5  4492.9 

7149.6  1251.8  1601.9  4574.2 

12 

14 

7073.8  1221.9  1553.2  4495.6 

7152.2  1252.9  1603.6  4576.9 

14 

16 

7076.4  1222.9  1554.9  4498.3 

7154.8  1253.9  1605.3  4579.7 

16 

18 

7079.0  1224.0-1556.5  4501.0 

7157.4  1255  0  1607.0  4582.4 

18 

20 

7081.7  1225.0  1558.2  4503.7 

7160.0  1256.0  1608.7  4585.1 

20 

22 

7084.3  1226.0  1559.9  4506.3 

7162.6  1257.0  1610.4  4587.9 

22 

24 

7086.9  1227.1  1561.5  4509.0 

71C5.2  1258.1  1612.1  4590.6 

24 

26 

7089.5  1228.1  1563.2  4511.7 

7167.8  1259.1  1613.8  4593.3 

26 

28 

7092.1  1229.1  1564.9  4514.4 

7170.4  1260.2  1615.5  4596.0 

28 

80 

7094.7  1230.2  1560.5  4517.1 

7173.0  1261.2  1617.3  45988 

30 

32 

7097.4  1231.2  1568.2  4519.8 

7175.6  1262.2  1619.0  4601.5 

32 

31 

7100.0  1232.2  1569.9  4522.5 

7178.2  1263.3  1620.7  4604.3 

34 

36 

7102.6  1233.3  1571.5  4525.3 

7180.8  1264.3  1622.4  4607.0 

36 

38 

7105.2  1234.8  1573.2  4528.0 

7183.4  1265.4  1624.1  4609.8 

38 

40 

7107.8  1235.3  1574.8  4530.7 

7186.0  1266.4  1625.9  4612.5 

40 

42 

7110.4  1236.4  1576.4  4533.4 

7188.6  1267.4  1627.6  4615.3 

4-> 

44 

7113.1  1237.4  1578.1  4536.1 

7191.2  1268.5  1629.3  4618.0 

44 

46 

7115.7  1238.4  1579.8  4538.8 

7193.8  1269.5  1631.0  4620.8 

40 

48 

7118.3  1239.5  1581.5  4541.5 

7196.4  1270.6  1632.7  4623.5 

48 

50 

7120.9  1240.5  1583.2  4544.2 

7199.0  3271.6  16345  4626.3 

50 

52 

7123.5  1241.5  1584.9  4547.0 

7201.6  1272.7  1636.2  4629.0 

52 

54 

7126.1  1242.6  1586.6  4549.7 

7204.2  1273.7  1637.9  4631.8 

54 

56 

7128.8  1243.6  1588.3  4552.4 

7206.8  1274.8  1639.6  4634.5 

56 

58 

7131.4  1244.6  1590.0  4555.1 

7209.4  1275.8  1641.3  4637.3 

58 

60 

7134.0  1245.6  1591.7  4557.8 

7212.0  1276.9  1643.1  4640.0 

60  / 

6  —FUNCTIONS  OF  A  ONE-DEGREE  CURVE.      151 


78° 

79° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

7212.0  1276.9  1643.1  4640.0 

7289.5  130S.5  1696.0  4723.4 

0 

o 

7214.6  1278.0  1644.8  4642.8 

7292.1  1309.5  1697.7  4726.2 

2 

4 

7217.2  1279.0  1646.6  4645.6 

7294.6  1310.6  1699.5  4729.0 

4 

6 

7219.7  1280.1  1648.3  4648.3 

7297.2  1311.7  1701.3  4731.8 

6 

8 

7222.3  1281.1  1650.1  4651.1 

7299.7  1312.7  1703.1  4734.7 

8 

10 

7224.9  1282.2  1651.8  4653.9 

7302.3  1313.8  1704.9  4737.5 

10 

12 

7227.5  1283.2  1653.6  4656.7 

7304.9  1314.9  1706.6  4740.3 

12 

14 

7230.1  1284.3  1655.3  4659.4 

7307.4  1315.9  1708.4  4743.1 

14 

16 

7232.7  1285.3  1657.1  4662.2 

7310.0  1317.0  1710.2  4745.9 

16 

18 

7235.2  1286.4  1G58.8  4665.0 

7312.6  1318.1  1712.0  4748.7 

18 

20 

7237.8  1287.4  1660.6  4667.7 

7315.1  1319.1  1713.8  4751.5 

20 

22 

7240.4  1288.5  1662.3  4670.5 

7317.7  1320*2  1715.6  4754.3 

22 

24 

7243.0  1289.5  1664.1  4673.3 

7320.3  1321.3  1717.4  4757.1 

24 

26 

7245.6  1290.6  1665.8  4676.0 

7322.8  1322.3  1719.2  4760.0 

26 

28 

7248.2  1291.6  1667.6  4678.8 

7325.4  1323.4  1721.0  4762.8 

28 

30 

7250.7  1292.7  1669.  3  4681.6 

7327.9  1324.5  1722.8  4765.6 

30 

32 

7253.3  1293.7  1671.1  4684.4 

7330.5  1325.5  1724.6  4768.4 

32 

34 

7255.9  1294.8  1672.8  4687.2 

7333.1  13:26.6  1726.4  4771.2 

34 

36 

7258.5  1295.8  1674.6  4689.9 

7335.6  1327.7  1728.2  4774.1 

36 

38 

7261.1  1296.9  1676.3  4692.7 

7338.2  1328.7  1730.0  4776.9 

38 

40 

7263.7  1297.9  1678.2  4695.5 

7340.8  1329.8  1731.9  4779.7 

"40 

42 

7266.2  1299.0  1679.9  4698  3 

7343.3  1330.8  1733.7  4782.6 

42 

44 

7268.8  1300.0.  1681.7  4701.1 

7345.9  1331.9  1735.5  4785.4 

44 

46 

7271.4  1301.1  1683.5  4703.9 

7348.4  1333.0  1737.3  4788.2 

46 

48 

7274.0  1302.1  1685.3  470''..  7 

7351.0  1334.1  1739.1  4791.0 

48 

50 

7276.6  1303.2  1687.1  4709.5 

7353.6  1335.2  1740.9  4793.9 

50 

52 

7279.2  1304.2  1688.8  4712.2 

7356.1  1336.2  1742.7  4796.7 

52 

54 

7281.7  1305.3  1690.6  4715.0 

7358.7  1337.3  1744.5  4799.5 

54 

56 

7284.3  1306.3  1692.4  4717.8 

7361.3  1338.4  1746.3  4802.4 

56 

58 

7286.9  13C7.4  1694.2  4720.6 

73o3.8  1339.5  1748.1  4805.2 

58 

60 

7289.5  1308.5  1696.0  4723.4 

73G6.4  1340.6  1750.0  4808.0 

60 

80° 

81° 

L.  C.   M.     E.     T. 

L.  C.   M.     E.     T. 

0 

7366.4  1340.6  1750  0  4808.0 

7442.7  1372.8  1805.5  4893.9 

0 

2 

7368  9  1341.7  1751.8  4810.9 

7445.2  1373.9  1807.3  4896.8 

2 

4 

7371.5  1342.7  1753.7  4813.7 

7447.7  1375.0  1809.2  4899.7 

4 

6 

7374.0  1343.8  1755.5  4816.6 

7450.3  1376.1  1811.1  4902.6 

6 

8 

7376.6  1344.9  1757.4  4819.4 

7452.8  1377.1  1813.0  4905.4 

8 

10 

7379.1  1346.0  1759.2  4822.3 

7455.3  1378.2  1814.9  4908.3 

10 

12 

7381.7  1347.0  1761.0  4825.1 

7457.8  1379.3  1816.8  4911.2 

12 

14 

7384.2  1348.1  1762.9  4828.0 

7460.4  1380.4  1818.6  4914.1 

14 

16 

7386.7  1349.2  1764.7  4830.8 

7462.9  1381.4  1820.5  4917.0 

16 

18 

7389.3  1350.3  1766.6  4833.7 

7465.4  1382.5  1822.4  4919.9 

•18 

20 

7391.8  1351.3  1768.4  4836.5 

7467.9  1383.6  1824.2  4922.8 

20' 

22 

7394.4  1352.4  1770.2  4839.4 

7470.4  1384.7  1826.1  4925.7 

22 

24 

7396.9  1353.5  1772.1  4842.2 

7473.0  1385.7  1828.0  4928.6 

24 

26 

7399.5  1354.6  1773.9  4845.1 

7475.5  1386.8  1829.9  4931.5 

26 

28 

7402.0  1355.6  1775.8  4847.9 

7478.0  1387.9  1831.8  4934.4 

28 

30 

7404.5  1356.7  1777.6  4850.8 

7480.5  1389.0  1833.7  4937.2 

30 

32 

7407.1  1357.8  1779.4  4853.7 

7483.1  1390.1  1835.6  4940.2 

82 

34 

7409.6  1358.9  1781.3  4856.5 

7485.6  1391.2  1837.5  4943.1 

34 

36 

7412.2  1359.9  1783.1  4859.4 

7488.1  1392.3  1839.4  4946.0 

36 

38 

7414.7  1361.0  1785.0  4862.3 

7490.6  1393.4  1841.3  4948.9 

38 

40 

7417.3  1362.1  1786.8  4865.1 

7493.2  1394.5  1843.2  4951.8 

40 

42 

7419.8  1363.2  1788.6  4868.0 

7495.7  1395.6  1845.1  4954.7 

42 

44 

7422.3  1364.2  1790.5  4870.9 

7498.2  1396.7  1847.0  4957.6 

44 

46 

7424.9  1365.3  1792.4  4873.8 

7500.7  1397.8  1848.9  4960.6 

46 

48 

7427.4  1366.4  1794  3  4876.6 

7503.3  1398.9  1850.8  4963.5 

48 

50 

7430.0  1367.5  1796.2  4879.5 

7505.8  1400.0  1852.7  4966.4 

60 

52 

7432.5  1368.5  1798.0  4882.4 

7508.3  1401.1  18546  4969.3 

52 

54 

7435.1  1369.6  1799.9  4885  3 

7510.8  1402.2  1856.5  4972.2 

54 

56 

7437.6  1370  7  1801.8  4888.1 

7513.3  1403.3  1858.4  4975.1 

56 

58 

7440.1  1371  .8  1803.7  4891.0 

7515.9  1404.4  1860.3  4978.0 

58 

60 

7442.7  1372.8  1805.5  4893.9 

7518.4  1405.5  18623  4981.  C 

60  \ 

152       6.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


82° 

83° 

L.  C.   M.    E.     T. 

L.  C.   M.     E.     T. 

0 

7518.4  1405.5  1862.3  4981.0 

7593.6  1438.5  1920.6  5069.4 

0 

2 

7520.9  1406.6  1§64.2  4983.9 

7596.1  1439.6  1922.6  5072.4 

2 

4 

7523.4  1407.7  1866.1  4986.8 

7598.6  1440.7  1924.6  5075.4 

4 

6 

7525.9  1408.8  1868.1  4989.8 

7601.1  1441.8  1926.5  5078.4 

6 

8 

7528.4  1409.9  1870.0  4992.7 

7603.6  1442.9  1928.5  5081.4 

8 

10 

7530.9  1411.0  1871.9  4995.7 

7606.0  1444.0  1930.5  5084.4 

10 

12 

7533.4  1412.1  1873  9  4998.6 

7608.5  1445.1  1932.4  5087.3 

12 

14 

7535.9  1413.2  1875.8  5001.5 

7611.0  1446.2  1934.4  5090.3 

14 

16 

7538.5  1414.3  1877.7  5004.5 

7613.5  1447.3  1936.4  5093.3 

16 

18 

7541.0  1415.4  1879.7  5007.4 

7616.0  1448.4  1938.4  5096.3 

18 

20 

7543.5  1416.5  1881.6  5010.3 

7618.5  1449.6  1940.4  5099.3 

20 

22 

7546.0  1417.6  1883.5  5013.3 

7621.0  1450.7  1942.4  5102.3 

22 

24 

7548.5  1418.7  1885.5  5016.2 

7623.5  1451.8  1944.4  5105.2 

24 

26 

7551.0  1419.8  1887.4  5019.2 

7626.0  1452.9  1946.4  5108.2 

26 

28 

7553.5  1420.9  1889.3  5022.1 

7628.5  1454.0  1948.4  5111.2 

28 

30 

7556.0  1422.0  1891.3  5025.0 

7630.9  1455.1  1950.4  5114.2 

30 

32 

7558.5  1423.1  1893.2  5028.0 

7633.4  1456.2  1952.4  5117.2 

32 

34 

7561.0  1424.2  1895.1  5031.0 

7635.9  1457.3  1954.4  5120.2 

34 

36 

7563.5  1425.3  1897.1  5033.9 

7638.4  1458.4  1956.4  5123.2 

36 

38 

7566.0  1426.4  1899.0  5036.9 

7640.9  1459.5  1958.4  5126.2 

38 

40* 

7568.5  1427.5  1901.0  5039.8 

7643.4  1460.7  1960.4  5129.2 

40 

42 

.7571.0  1428.6  1902.9  5042.8 

7645.9  1461.8  1962.4  5132.2 

42 

44 

7573.5  1429.7  1904.9  5045.8 

7648.4  1462.9  1964.4  5135.2 

44 

46 

7576.1  1430.8  1906.9  5048.7 

7650.9  1464.0  1966.4  5138.2 

46 

48 

7578.6  1431.9  1908.8  5051.7 

7653.4  1465.1  1968.4  5141.2 

48 

50 

7581.1  1433.0  1910.8  5054.6 

7655.8  1466.2  1970.4  5144.3 

50 

52 

7583.6  1434.1  1912.8  5057.6 

7658.3  1467.3  1972.4  5147.3 

52 

54 

7586.1  1435.2  1914.7  5060.6 

7660.8  1468.4  1974.4  5150.3 

54 

56 

7588.6  1436.3  1916.7  5063.5 

7663.3  1469.5  1876.4  5153.3 

56 

58 

7591.1  1437.4  1918.7  5066.5 

7665.8  1470.6"  1978.4  5156.3 

58 

60 

7593.6  1438.5  1920.6  5069.4 

7668.3  1471.8  1980.5  5159.3 

60 

84° 

85° 

t 

L.  C.   M.    E.     T. 

L.  C.    M.    E.    1 

0 

7668.3  1471.8  1980.5  5159.3 

7742.4  1505.4  2041.8  5250.6 

0 

2 

7670.8  1472.9  1982.5  5162.3 

7744.8  1506.5  2043.9  5253.6 

2 

4 

7673.2  1474.0  1984.5  5165.3 

7747.3  1507.6  2046.0  5256.7 

4 

6 

7675.7  1475.1  1986.6  5168.4 

7749.7  1508.8  2048.0  5259.8 

6 

8 

7678.2  1476.2  1988.6  5171.4 

7752.2  1509.9  2050.1  5262.9 

8 

10 

7680.6  1477.4  1990.6  5174.4 

7754.6  1511.0  2052.2  5266  0 

10 

12 

7683.1  1478.5  1992.7  5177.5 

7757.1  1512.2  2054.2  5269.0 

12 

14 

7685.6  1479.6  1994.7  5180.5 

7759.5  1513.3  2056.3  5272.1 

14 

16 

7688.1  1480.7  1996.7  5183.5 

7762.0  1514.4  2058.4  5275.2 

16 

18 

7690.5  1481.8  1998.8  5186.6 

7764.4  1515.6  2060.5  5278.3 

18 

"20 

7693.0  1483.0  2000.8  5189.6 

7766.9  1516.7  2062.6  5281.4 

20 

22 

7695.5  1484.1  2002.8  5192.6 

7769.3  1517.8  2064.7  5284.4 

22 

24 

7697.9  1485.2  2004.9  5195.6 

7771.8  1519.0  2066.8  5287.5 

24 

26 

7700.4  1486.3  2006.9  5198.7 

7774.2  1520.1  2068.9  5290.6 

26 

28 

7702.9  1487.4  2008.9  5201.7 

7776.7  1521.2  2071.0  5293.7 

28 

30 

7705.3  1488.6  2011.0  5204.7 

7779.1  1522.4  2073.1  5296.7 

30 

32 

7707.8  1489.7  2013.0  5207.8 

7781.5  1523.5  2075.2  5299.8 

32 

84 

7710.3  1490.8  2015.0  5210.8 

7784.0  1524.6  2077  3  5302.9 

34 

36 

7712.8  1491.9  2017.0  5213.9 

7786.4  1525.8  2079.4  5306.1 

36 

38 

7715.2  1493.0  2019.1  5216.9 

7788.9  1526.9  2081.5  5309.2 

38 

40 

7717.7  1494.2  2021.2  5220.0 

7791.3  1528.0  2083.7  5312.3 

40 

42 

7720.2  1495.3  2023.2  5223.1 

7793.8  1529.2  2085.8  5315.4 

42 

44 

7722.6  1496.4  2025.3  5226.1 

7796.2  1530.3  2087.9  5318.5 

44 

46 

7725.1  1497.5  2027.4  5229.2 

7798.7  1531.4  2090.0  5321.6 

46 

48 

7727.6  1498.6  2029.4  5232.2 

7801.1  1532.6  2092.1  5324.7 

48 

50 

7730.0  1499.8  2031.5  5235.3 

7803.6  1533.7  2094.2  5327.8 

50 

52 

7732.5  1500.9  2033.6  5238.3 

7806.0  1534.8  2096.3  5330  9 

52 

54 

7735.0  1502.0  2035.6  5241  4 

7808.5  1536.0  2098.4  5334.0 

54 

56 

7737.5  1503.1  2037.7  5244.5 

7810.9  1537.1  21.00.6  5337.1 

56 

58 

7739  9  1504.2  2039.8  5247.5 

7813  4  1538.2  2102.7  5340.2 

58 

60 

7742.4  15054  2041.8  5250.6 

7815.8  1539.3  2104.8  5343.3 

60 

6.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE.     153 


86° 

87° 

L.  C.   M.    E.    ,T. 

L.  C.   M.    E.    T, 

/ 

0 

7815.8  1539.3  2104.8  5343.3 

7888.5  1573.6  2169.5  5437.5 

b 

2 

7818.2  1540.4  2106.9  5346.4 

7890.9  1574.8  2171.6  5440.7 

2 

4 

7820.6  1541.6  2109.1  5349.5 

7893.3  1575.9  2173.8  5443.9 

4 

6 

7823.1  1542.7  2111.2  5352.7 

7895.7  1577.1  2176.0  5447.1 

6 

8 

7825.5  1543.9  2113.4  5355.8 

7898.1  1578.2  2178.2  5450.3 

8 

10 

7827.9  1545.0  2115.5  5358.9 

7900.5  1579.4  2180.4  5453.4 

10 

12 

7830.3  1546.1  2117.6  5362.0 

7903.0  1580.5  2182.5  5456.6 

12 

14 

7832.8  1547.3  2119.8  5365.2 

7905.4  1581.7  2184.7  5159.8 

14 

16 

7835.2  1548.4  2121.9  5368.3 

7907.8  1582.9  2186.9  5463.0 

16 

18 

7837.6  1549.6  2124.1  5371.4 

7910.2  1584.0  2189.1  5466.2 

18 

20 

7840.0  1550.7  2126.2  5374.6 

7912.6  1585.1  2191.3  5469.4 

20 

22 

7842.4  1551.8  2128.3  5377.7 

7915.0  1586.3  2193.5  5472.5 

22 

24 

7844.9  1553.0  2130.5  5r80.8 

7917.4  1587.4  2195.7  5475.7 

24 

26 

7847.3  1554.1  2132.6  5383.9 

7919.8  1588.6  2197.9  5478.9 

26 

28 

7849.7  1555.3  2134.8  5387.1 

7922.2  1589.7  2200.1  5482.1 

28 

30 

7852.1  1556.4  2136.9  5390.2 

7924.6  1590.9  2202.3  5485.3 

30 

32 

7854.6  1557.5  2139.0  5393.4 

7927.1  1592.0  2204.5  5488.5 

82 

34 

7857.0  1558.7  2141.2  5396.5 

7929.5  1593.2  2206.8  5491.7 

34 

36 

7859.4  1559.8  2143.3  5399.7 

7931.9  1594.3  2209.0  5494.9 

36 

38 

7861.8  1561.0  2145.5  5402.8 

7934.3  1595.5  2211.2  5498.1 

38 

40 

7864.3  1562.1  2147.7  5406.0 

7936.7  1596.6  2213.4  5501.3 

40 

42 

7866.7  1563.2.2149.8  5409.1 

7939.1  1597.8  2215.6  5504.5 

42 

44 

7869.1  1564.4  2152.0  5412.3 

7941.5  1598.9  2217.8  5507.7 

44 

46 

7871.5  1565.5  2154.2  5415.4 

7913.9  1600.1  2220.0  5510.9 

46 

48 

7874.0  1566.7  2156.4  5418.6 

7946.3  1601.2  2222.3  5514.1 

48 

50 

7876.4  1567.8  2158.6  5421.8 

7948.7  1602.4  2224.5  5517.3 

50 

52 

7878.8  1568.9  2160.7  5424.9 

7951.2  1603.5  2226.7  5520.5 

52 

54 

7881.2  1570.1  2162.9  5428.1 

7953.6  1604.7  2228.9  5523.7 

54 

56 

7883.6  1571.2  2165.1  5431.2 

7956.0  1605.8  2231.1  5526.9 

56 

58 

7886.1  1572.4  2167.3  5434.4 

7958.4  1607.0  2233.3  5530.1 

58 

60 

7888.5  1573.6  2169.5  5437.5 

7960.8  1608.2  2235.6  5533.3 

60 

88° 

89° 

L.  C.   M.    E.     T. 

L.  C.   M.    E.    T. 

0 

7960.8  1608.2  2235.6  5533.3 

8032.4  1643.0  2303.6  5630.8 

0 

2 

7963.2  1609.4  2237.8  5536.6 

8034.8  1644.1  2305.9  5631.1 

2 

4 

7965.6  1610.5  2240.1  5539.8 

8037.1  1645.3  2308.2  5637.4 

4 

6 

7968.0  1611.7  2242.3  5543.1 

8039.5  1646.5  2310.5  5640.7 

6 

8 

7970.3  1612.8  2244.6  5546.3 

8041.9  1647.7  2312.8  5644.0 

8 

10 

7972.7  1614.0  2246.8  5549.5 

8044.2  1648.9  2315.1  5647.3 

10 

1? 

7975.1  1615.2  2249.1  .5552.8 

8046.6  1650.0  2317.4  5650.6 

12 

14 

7977.5  1616.3  2251.3  5556.0 

8049.0  1651.2  2319.7  5653.9 

14 

16 

7979.9  1617.5  2253.6  5559.2 

8051.4  1652.4  2322.0  5657.1 

16 

Ib 

7982.3  1618.6  2255.8  5562.5 

8053.7  1653.6  2324.3  5660.4 

18 

20 

7984.7  1619.8  2258.1  5565.7 

8056.1  1654.8  2326.7  5663.7 

20 

22 

7987.1  1621.0  2260.4  5568.9 

8058.5  1655.9  2329.0  5667.0 

22 

24 

7989.4  1622.1  2262.7  5572.2 

8060.8  1657.1  2331.3  5670.3 

24 

26 

7991.8  1623.3  2264.9  5575.4 

8063.2  1658.3  2333.7  5673.6 

26 

28 

7994.2  1624.4  2267.2  5578.6 

8065.6  1659.5  2336.0  5676.9 

28 

30 

7996.6  1625.6  2269.5  5581.9 

8067.9  1660.7  2338.3  5680.2 

30 

32 

7999.0  1626.8  2271.7  5585.1 

8070.3  1661.8  2340.7  5683.5 

32 

34 

8001.4  1627.9  2273.9  5588.4 

8073.7  1663.0  2343.0  5686.8 

34 

36 

8003.8  1629.1  2276.2  5591.7 

8075.1  1664.2  2345.3  5690.2 

36 

38 

8006.1  1630.2  2278.5  5594.9 

8077.4  1665.4  2347.7  5693.5 

38 

40 

8008.5  1631.4  2280.8  5598.2 

8079.8  1666.6  2350.0  5696.8 

40 

42 

8010.9  1632.6  2283.0  5601.4 

8082.2  1667.7  2352.3  5700.1 

42 

44 

8013.3  1633.7  2285.3  5604.7 

8084.5  1668.8  2354.7  5703.4 

44 

46 

8015.7  1634.9  2287.6  5608.0 

8086.9  1670.0  2357.0  5706.8 

46 

48 

8018.1  1636.0  2289.9  5611.2 

8089.3  1671.2  2359.3  5710.1 

48 

50 

8020.5  1637.2  2292.2  5614.5 

8091.6  1672.4  2361.7  5713.4 

50 

52 

8022.9  1638.4  2294.4  5617.8 

8094.0  1673.5  2364.0  5716.7 

52 

54 

8025.2  1639.5  2296.7  5621.0 

8096.4  1674.7  2366.3  5720.0 

54 

56 

8027.6  1640.7  2299.0  5624.3 

8098.8  1675.9  2368.7  5723.4 

56 

58 

8030.0  1641.8  2301.3  56-37.5 

8101.1  1677.1  2371.0  5726.7 

58 

60 

8032.4  1643.0  2303.6  5630.8 

8103.5  1678.3  2373.4  5730.0 

60 

154       6.— FUNCTIONS  OF  A  ONE-DEGREE  CURVE. 


90° 

91° 

L.  C.   M.    E.     T. 

L.  C.   M.    E.     T. 

0 

8103.5  1678.3  2373.4  5730.0 

8173.9   713.8  2445.1  5830.9 

0 

2 

8105.8  1679.5  2375.8  5733.3 

8176.2   715.0  2447.5  5834.3 

2 

4 

8108.2  1680.6  2378.2  5736.7 

8178.5   716.2  2450.0  5837.7 

4 

6 

8110.5  1681.8  2380.5  5740.0 

8180.9   717.4  2452.4  5841.1 

6 

8 

8112.9  1683.0  2382.9  5743.4 

8183.2   718.6  2454.8  5844.5 

8 

10 

8115.2  1684.2  2385.3  5746.7 

8185.5   719.7  2457.2  5847.9 

10 

12 

81176  1685.4  2387.6  5750.0 

8187.9   720.9  2459.7  5851.3 

12 

14 

8119.9  1686.5  2390.0  5753.4 

8190.2  1722.1  2462.1  5854.7 

14 

16 

8122.3  1687.7  2392.4  5756.7 

8192.5  1723.3  2464.5  5858.1 

16 

18 

8124.6  1688.9  2394.7  5760.1 

8194.8  1724.5  2467.0  5861.5 

18 

20 

8127.0  1690.1  2397.1  5763.4 

8197.2  1725.7  2469.4  5864.9 

20 

22 

8129.3  1691.3  2399.5  5766.8 

8199.5  1726.9  2471.9  5868.3 

22 

24 

8131.7  1692.5  2401.9  5770.1 

8201.8  1728.1  2474.3  5871.8 

24 

26 

8134.0  1693.6  2404.3  5773.5 

8204.2  1729.3  2476.7  5875.2 

26 

28 

8136.4  1694.8  2406.6  5776.9 

8206.5  1730.5  2479.2  5878.6 

28 

30 

8138.7  1696.0  2409.0  5780.2 

8208.8  1731.7  2481.6  5882.0 

30 

32 

8141.1  1697.2  2411.4  5783.6 

8211.1  1732.9  2484.1  5885.4 

32 

34 

8143.4  1698.4  2413.8  5787.0 

8213.5  1734.1  2486.5  5888.9 

34 

36 

8145.8  1699.6  2416.2  5790.3 

8215.8  1735.3  2489.0  5892.3 

36 

38 

8148.1  1700.7  2418.6  5793.7 

8218.1  1736.4  2491.5  5895.7 

38 

40 

8150.4  1701.9  2421.0  5797.1 

8220.4  1737.6  2493.9  5899.2 

40 

42 

8152.8  1703.1  2423.4  5800.4 

8222.8  1738.8  2496.4  5902.6 

42 

44 

8155.1  1704.3  2425.8  5803.8 

8225.1  1740.0  2498.9  5906.0 

44 

46 

8157.5  1705.5  2428.2  5807.2 

8227.4  1741.2  2501.3  5909.4 

46 

48 

8159.8  1706.7  2430.6  5810.6 

8229.7  1742.4  2503.8  5912.9 

48 

50 

8162.2  1707.9  2433.0  5814.0 

8232.0  1743.6  2506.3  5916.3 

50 

52 

8164.5  1709.0  2435.4  5817.3 

8234.3  1744.8  2508.7  5919.8 

52 

54 

8166.8  1710.2  2437.9  58:20.7 

8236.7  1746.0  2511.2  5923.2 

54 

56 

81G9.2  1711.4  2440.3  5824.1 

8239.0  1747.2  2513.7  5926.7 

56 

58 

8171.5  1712.6  2442.7  5827.5 

8241.3  1748.4  2516.2  5930.1 

58 

60 

8173.9  1713.8  2445.1  5830.9 

8243.6  1749.6  2518.7  5933.6 

60 

92° 

93° 

/ 

L.  C    M.    E.     T. 

L.  C.   M.    E.     T. 

0 

8243.6  1749.6  2518.7  5933.6 

8312.8  1785.7  2594.2  603S.2 

0 

2 

8245.9  1750.8  2521.2  5937.0 

8315.1  1786.9  2596.8  6041.7 

2 

4 

8248.2  1752.0  2523.6  5940.5 

8317.4  1188.2  2599.3  6045.2 

4 

6 

8250.6  1753.2  2526.1  5944.0 

8319.7  1789.4  2G01.9  6048.7 

6 

\ 

8252.9  1754.4  2528.6  5947.4 

8322.0  1790.6  2604.4  6052.2 

8 

10 

8255.2  1755.6  2531.1  5950.9 

8324.3  1791.8  2G07.0  6055.8 

10 

12 

8257.5  1756.8  2533.6  5954.4 

8326.6  1793.0  2609.6  6059.3 

12 

14 

8259.8  1758.0  2536.1  5957.8 

8328.8  1794.2  2612.1  6062.8 

14 

16 

8262.2  1759.2  2538.6  5961.3 

8331.1  1795.4  2614.7  6066.4 

16 

18 

8264.5  1760.4  2541.1  5964.8 

8333.3  1796.6  2617.3  6069.9 

18 

20 

8266.8  1761.6  2543.6  5968.2 

8335.6  1797.8  2619.8  6073.4 

20 

22 

8269.1  17G2.8  2546.1  5971.7 

8337.9  1799.1  2622.4  6077.0 

22 

24 

8271.4  1764.0  2548.6  5975.2 

8340.2  1800.3  2625.0  6080.5 

24 

26 

8273.7  1765.2  2551.2  5978.7 

8342.5  1801.5  2627.6  6084.1 

26 

28 

8276.0  1766.4  2553.7  5982.2 

8344.8  1802.7  2630.2  6087.6 

28 

30 

8278.3  1767.6  2556.2  5985.6 

8347.1  1803.9  2632.7  6091.2 

30 

32 

8280.6  1768.8  2558.7  5989.1 

8349.4  1805.1  2635.3  6094.7 

32 

34 

8282.9  17^0.0  2561.2  5992.6 

8351.7  1806.3  2637.9  6098.3 

34 

36 

8285.2  1771.2  2563.8  5996.1 

8354.0  1807.6  2640.5  6101.8 

36 

.38 

8287.5  1772.5  2566.3  5999.6 

8356.3  1808.8  2643.1  6105.4 

38 

40 

8289.8  1773.7  2568.8  6003.1 

8358.5  1810.0  2645.7  6109.0 

40 

42 

8292.1  1774.9  2571.3  6006.6 

8360.8  1811.2  2648.3  6112.5 

42 

44 

8294.4  1776.1  2573.9  6010.1 

8363.1  1812.4  2650.9  6116.1 

44 

46 

8296.7  1777.3  2576.4  6013.6 

8365.4  1813.6  2653.5  6119.7 

46 

48 

8299.0  1778.5  2578.9  6017.1 

8367.7  1814.9  2656.1  6123.2 

48 

50 

8301.3  1779.7  2581.5  6020.6 

8369.9  1816.1  2658.7  6126.8 

50 

52 

8303.6  1780.9  2584.0  6024.1 

8372.2  1817.3  2661.3  6130.4 

52 

54 

8305.9  1782.1  2586.6  6027.6 

8374.5  1818.5  2663.9  6133.9 

54 

56 

8308.2  1783.3  2589.1  6031.1 

8376.8  1819.7  2666.6  6137.5 

56 

58 

8310.5  1784.5  2591.7  6034.6 

8379.1  1820.9  2669.2  6141.1 

58 

GO 

8312.8  1785.7  2594.2  6038.2 

8381.3  1822.2  2671.8  6144.7 

60 

6  —FUNCTIONS  OF  A  ONE-DEGREE  CURVE.    155 


94° 

95° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.     T. 

0 

8381.3  1822.2  2671.8  6144.7 

8449.2  1858.9  2751.5  6253.2 

0 

2 

8383.6  1823.4  2674.4  6148.3 

8451.5  1860.1  2754.2  6256.9 

2 

4 

8385.9  1824.6  2677.0  6151.9 

8453.7  1861.3  2756.9  6260.5 

4 

6 

8388.1  1825.8  2679.7  6155.4 

8456.0  1862.6  2759.6  6264.2 

6 

8 

8390.4  1827.0  2682.3  6159.0 

8458.2  1863.8  2762.3  6267.8 

8 

10 

8392.7  1828.3  2684.9  6162.6 

8460.4  1865.0  27650  6271.5 

10 

12 

8395.0  1829.5  2687.6  6166.2 

8462.7  1866.3  2767.7  6275.2 

12 

14 

8397.2  1830.7  2690.2  6169.8 

8464.9  1867.5  2770.4  6278.8 

14 

16 

8399.5  1831.3  2692.8  6173.4 

8467.2  1868.7  2773.1  6282.5 

16 

18 

8401.7  1833.1  2695.6  6177.0 

8469.4  1869.9  2775.8  6286.2 

18 

20 

8404.  C  1834.4  2698.1  6180.6 

8471.7  1871.2  2778.5  6289.8 

20 

22 

8406.3  1835.6  2700.8  6184.2 

8473.9  1872.4  2781.2  6293.5 

22 

24 

8408.5  1836.8  2703.4  6187.8 

8476.2  1873.6  2784.0  6297.2 

24 

26 

8410.8  1838.0  2706.1  6191.5 

8478.4  1874.9  2786.7  6300.9 

26 

28 

8413.1  18J9.3  2708.7  6195.1 

8480.7  1876.1  2789.4  6304.6 

28 

30 

8415.3  1840.5  2711.4  6198.7 

8482.9  1877.3  2792.1  6308  2 

30 

32 

8417.6  1841.7  2714.0  6202.3 

8485.1  1878.6  2794.9  6311.9 

32 

34 

8419.9  1842.9  2716.7  6205.9 

8487.4  1879.8  2797.6  6315.6 

34 

36 

8422.1  1844.2  2719.3  6209.5 

8489.6  1881.0  280U.3  6319.3 

36 

38 

8424.4  1845.4  2722.0  6213.2 

8491.9  1882.3  2803.1  6323.0 

38 

40 

8426.6  1846.6  2724.7  6216.8 

8494.1  1883.5  2805.8  6326.7 

40 

42 

8428.9  1847.8  itf27.3  6220.4 

8496.3  1884.8  2808.6  6330.4 

42 

44 

8431.2  1849.1  2730.0  6224.1 

8498.6  1886.0  2811.3  6334.1 

44 

46 

8433.4  1850.3  2732.7  6227.7 

8500.8  1887.2  2814.1  6337.8 

46 

48 

8435.7  1851.5  2735.4  6231.3 

8503.0  1888.5  2816.8  6341.5 

48 

50 

8437.9  1852.7  2738.0  6235.0 

8505.3  1889.7  2819.6  6345.2 

50 

52 

8440.2  1854.0  2740.7  6238.6 

8507.5  1890.9  2822.3  6349.0 

52 

54 

8442.4  1855.2  2743.4  6242.3 

8509.8  1892.2  2825.1  6352.7 

54 

56 

8444.7  1856.4  2746.1  6245.9 

8512.0  1893.4  2827.8  6356.4 

56 

58 

8447.0  1857.6  2748.8  6249.6 

8514.2  1894.6  2830.6  6360.1 

58 

60 

8449.2  1858.9  2751.5  6253.2 

8516.4  1895.9  2833.4  6363.8 

60 

96° 

97° 

/ 

L.  C.    M.    E.     T. 

L.  C.    M.    E.     T. 

t 

0 

8516.4  1895.  S  2833.4  6363.8 

8583.0  1933.2  2917.5  6476.6 

0 

2 

8518.7  1897.1  2836.1  6367.5 

8585.2  1934.4  2900.3  6480.4 

2 

4 

8520.9  1898.4  2838.9  6371.3 

8587.5  1935.7  2923.2  6484.2 

4 

6 

8523.1  1899.6  2841.7  6375.0 

8589.7  1936.9  2926.0  6488.0 

6 

8 

8525.4  1900.8  2844.5  6378.7 

8591.9  1938.2  2928.9  6491.8 

8 

10 

8527.6  1902.1  2847.2  6382.5 

8594.1  1939.4  2931.7  6495.6 

10 

12 

8529.8  1903.3  2850.0  6386.2 

8596.3  1940.7  2934.6  6499.4 

12 

14 

8532.0  19046  2852.8  6389.9 

8598.5  1941.9  2937.5  6503  2 

14 

16 

8534.3  1905.8  2855.6  6393.7 

8600.7  1943.2  2940.3  6507.1 

16 

18 

8536.5  1907.0  2858.4  6397.4 

8602.9  1944.4  2943.2  6510.9 

18 

20 

8538.7  1908.3  2861.2  6401.2 

8605.1  1945.7  2946.1  6514.7 

20 

22 

8540.9  1909.5  2864.0  6404.9 

8607.3  1946.9  2948.9  6518.5 

22 

24 

8543.2  1910.8  2866.7  6408.7 

8609.5  1948.2  2951.8  6522.3 

24 

26 

8545.4  1912.0  2869.5  6412.4 

8611.7  1949.4  2954.7  6526.2 

26 

28 

8547.  e  1913.3  2872.3  6416.2 

8613.9  1950.7  2957.6  6530.0 

28 

30 

8549.8  1914.5  2875.1  6419.9 

8616.1  1952:0  29G0.4  6533.8 

30 

32 

8652.0  1915.7  2877.9  6423.7 

8618.3  1953.2  2963.3  6537.7 

32 

34 

8554.3  1917.0  2880.8  6427.5 

8620.5  1954.5  2966.2  6541.5 

'  34 

36 

8556.5  1918.2  2883.6  6431.2 

8622.7  1955.7  2969.1  6545.3 

36 

38 

8558.7  1919.5  2886.4  6435.0 

8624.9  1957.0  2972.0  6549.2 

38 

40 

8560.9  1920.7  2889.2  6438.8 

8627.1  1958.2  2974.9  6553.0 

40 

42 

8563.1  1922.0  2892.0  6442  5 

8629.3  1959  5  2977.8  6556  9 

42 

44 

8565  3  1923.2  2894.8  6446.3 

8631.5  1960.7  2980.7  6560.7 

44 

46 

8567  6  1924  5  2897.7  6450.1 

8633.7  1962.0  2983.6  6564.6 

46 

48 

8569  8  1925.7  2900.5  6453.9 

8635.8  1963.2  2986.5  6568  4 

48 

50 

8572.0  1927.0  2903.3  6457.6 

8638.0  1964.5  2989.4  6572.3 

50 

52 

8574.2  1928.2  2906.1  6461.4 

8040.2  1965.8  2992.3  6576.2 

52 

54 

8576.4  1929  4  2W9.0  6465.2 

8642.4  1967.0  2995.2  6580.0 

54 

56 

8578.6  1930.7  2911.8  6-169.0 

8644.6  1968.3  2998.1  6583.9 

56 

58 

8580.8  1931.9  2914.7  6472.8 

8646.8  1969.5  3001.1  6587.7 

58 

60 

8583.0  1933.2  2917.5  6476.6 

8649  0  1970.8  3004  0  6591.6 

60 

156    6.— FUNCTIONS  OF    A.  ONE-DEGREE   CURVE. 


98° 

99° 

L.  C.    M.     B.    T. 

L.  C.    M.    E.    T. 

0 

8649.0  1970.8  3004.0  6591.6 

8714.3  2008.7  3092.9  6709.0 

0 

2 

8651.2  1972.0  3006.9  6595.5 

8716.4  2009.9  3095.9  6712.9 

2 

4 

8653.3  1973.3  3009.8  6599.4 

8718.6  2011.2  3098.9  6716.9 

4 

6 

8655.5  1974.6  3012.8  6603.2 

8720.7  2012.5  3101.9  6720.8 

6 

8 

8657.7  1975.8  3015.7  6607.1 

8722.9  2013.7  3104.9  6724.8 

8 

10 

8659.9  1977.1  3018.6  6611.0 

8725.1  2015.0  3107.9  6728.8 

10 

12 

8662.1  1978.3  3021.6  6614.9 

8727.2  2016.3  3111.0  6732.7 

12 

14 

8664.3  1979.6  3024.5  6618.8 

8729.4  2017.5  3114.0  6736.7 

14 

16 

8666.4  1980.9  3027.5  6622.7 

8731.5  2018.8  3117.0  6740.7 

16 

18 

8668.6  1982.1  3030.4  6626.6 

8733.7  2020.1  3120.0  6744.6 

18 

20 

8670.8  1983.4  3033.3  6630.5 

8735.9  2021.4  3123.1  6748.6 

20 

22 

8673.0  1984.6  3036.3  6634.4 

8738.0  2022.6  3126.1  6752.6 

22 

24 

8675.2  1985.9  3039.3  6638.3 

8710.2  2023.9  3129.1  6756.6 

24 

26 

8677.8  1987.2  3042.2  6642.2 

8742.3  2025.2  3132.2  6760.6 

26 

28 

8679.5  1988.4  3045.2  6646.1 

8744.5  2026.4  3135.2  G764.6 

28 

30 

8681.7  1989.7  3048.1  6650.0 

8746.6  2027.7  3138.3  6768.6 

30 

32 

8683.9  1991.0  3051.1  6653.9 

8748.8  2029.0  3141.3  6772.6 

32 

34 

8686.0  1992.2  3054.1  6657.8 

8750.9  2030.3  3144.4  6776.6 

34 

36 

8688.2  1993.5  3057.0  6661.7 

8753.1  2031.5  3147.4  6780.6 

36 

38 

8690.4  1994.7  3060.0  6665.7 

8755.3  2032.8  3150.5  6784.6 

38 

40 

8692.6  1996.0  3063.0  6669.6 

8757.4  2034.1  3153.5  6788.6 

40 

42 

8694.7  1997.3  3066.0  6673.5 

8759.5  2035.4  3156.6  6792.6 

42 

44 

8696.9  1998.5  3068.9  6677.4 

8761.7  2036.6  3159.7  6796.6 

44 

46 

8699.1  1999.8  3071.9  6681.4 

8763.8  2037.9  3162.7  6800.6 

46 

48 

8701.2  2001.1  3074.9  6685.3 

8766.0  2039.2  3165.8  6804.6 

48 

60 

8703.4  2002.3  3077.9  6689.2 

8768.1  2040.5  3168.9  6808.6 

50 

52 

8705.6  2003.6  3080.9  6693.2 

8770.3  2041.7  3172.0  6812.6 

52 

54 

8707.8  2004.9  3083.9  6697.1 

8772.4  2043.0  3175.1  6816.7 

54 

56 

8709.9  2006.1  3086.9  6701.1 

8774.6  2044.3  3178.1  6820.7 

56 

58 

8712.1  2007.4  3089.9  6705.2 

8776.7  2045.6  3181.2  6824.7 

58 

60 

8714.3  2008.7  3092.9  6709.0 

8778.9  2046.8  3184.3  6828.8 

60 

100° 

101° 

L  C.    M.     E.    T. 

L.  C.   M.    E.    T. 

/ 

" 

8778.9  2046.8  3184.3  6828.8 

8842.8  2085.3  3278.3  6951  0 

0 

2 

8781.0  2048.1  3187.4  6832.8 

8844.9  2086.6  3281.5  6955.2 

2 

4 

8783.1  2049.4  3190.5  6836.8 

8847.0  2087.8  3284.7  6959.3 

4 

6 

8785.3  2050.7  3193.6  6840.9 

8849.2  2089.1  3287.9  6963.4 

6 

8 

8787.4  2051.9  3196.7  6844.9 

8851.3  2090.4  3291.1  6967.6 

8 

10 

8789.6  2053.2  3199.8  6,849.0 

8853.4  2091.7  3294.3  6971.7 

10 

12 

8791.7  2054.5  3:202.9  6853.0 

8855.5  2093.0  3297.5  6975.8 

12 

14 

8793.9  2055.8  3:206.0  6857.1 

8857.6  2094.3  3300.7  6980.0 

14 

16 

8796.0  2057.1  3209.1  6861.1 

8859.8  2095.6  3303.9  6984  1 

16 

18 

8798.9  2058.3  3212.2  6865.2 

8861.9  2096.9  3307.1  6988.2 

18 

20 

8800.3  2059.6  3215.4  6869.2 

8864.0  2098.2  3310.3  6992.4 

20 

22 

8802.4  2060.9  3218.5  6873.3 

8866.1  2099.4  3313.5  6996.6 

22 

24 

8804.5  2062.2  3221.6  6877.4 

8868.2  2100.7  3316.7  7000.7 

24 

26 

8806.7  2063.5  3224.7  6881.4 

8870.3  2102.0  3319.9  7004.9 

26 

28 

8808.8  2064.7  3227.9  6885.5 

8872.4  2103.3  3323.1  7009.0 

28 

30 

8810.9  2066.0  3231  0  6889.6 

8874.5  2104.6  3326.4  7013.2 

30 

32 

8813.1  2067.3  3234.1  6893.7 

8876.7  2105.9  3329.6  7017.3 

32 

34 

8815.2  2068.6  3237  3  6897.8 

8878.8  2107.2  3332.8  7021.5 

34 

36 

8817.3  2069.9  32404  6901.8 

8880.9  2108.5  3336.0  7025.7 

36 

38 

8819.5  2071.1  3243.5  6905.9 

8883.0  2109.8  3339.3  7029.9 

38 

40 

8821.6  2072.4  3246.7  6910.0 

8885.1  2111.1  3342.5  7034.0 

40 

42  ' 

8823.7  2073.7  3249.8  6914.1 

8887.2  2112.4  3345.8  7038.2 

42 

44 

8825.8  2075.0  3253.0  6918.2 

8889.3  2113.6  3349.0  7042.4 

44 

46 

8828.0  2076.3  3256.2  6922.3 

8891.4  2114.9  3352.3  7046.6 

46 

48 

8830.1  2077.6  3259.3  6926.4 

8893.5  2116.2  3355.5  7050.8 

48 

50 

8832.2  2078.9  3262.5  6930.5 

8895.6  2117.5  3358.8  7055.0 

50 

52 

8834.3  2080.1  3265.7  6934.6 

8897.7  2118.8  3362.0  7059.2 

52 

54 

8836.4  2081.4  3268.8  6938.7 

8899.8  2120.1  3365.5  70634 

54 

56 

8838.6  2082.7  3272.0  6942.8 

8901.9  2121.4  3368.7  7067.6 

56 

58 

8340.7  2084.0  3275.2  6946.9 

8904.0  2122.7  3372.0  7071.8 

58 

60 

88428  2085.3  3278.3  6951.0 

8906.1  2124.0  3375.1  7076.0 

60 

6.— FUNCTIONS   OF  A  ONE-DEGREE   CURVE.     157 


102° 

103° 

L.  C.   M.    E.     T. 

L.  C.   M.    E.     T. 

0 

8906.1  2124.0  3375.1  7076.0 

8968.7  2163.0  3474.6  7203.6 

0 

2 

8908.2  2125.3  3378.3  7080.2 

8970.8  2164.3  3478.0  7207.9 

2 

4 

8910.3  2126.6  3381.6  7084.4 

8972.9  2165.6  3481.4  7212.2 

4 

6 

8912.4  2127.9  3384.9  7088.6 

8974.9  2166.9  3484.7  7216.5 

6 

8 

8914.5  2129.2  3388.2  7092.8 

8977.0  2168.2  3488.1  7220.8 

8 

10 

8916.6  2130.5  3391.5  7097.1 

8979.1  2169.5  3491.5  7225.1 

10 

12 

8918.7  2131.8  3394.7  7101.3 

8981.1  2170.8  3494.9  7229.5 

12 

14 

8920.8  2133.1  3398.0  7105.5 

8983.2  2172.1  3498.3  7233.8 

14 

16 

8922.9  2134.4  3401.3  7109.7 

8985.3  2173.4  3501.6  7238.1 

16 

18 

8925.0  2135.7  3404.6  7114.0 

8987.3  2174.7  3505.3  7242.4 

18 

20 

8927.0  2137.0  3407.9  7118.2 

8989.4  2176.1  3508.4  7246.8 

20 

22 

8929.1  2138.3  3411.2  7122.4 

8991.5  2177.4  3511.8  7251.1 

30 

24 

8931.2  2139.6  3414.5  7126.7 

8993.5  2178.7  3515.2  7255.4 

24 

26 

8933.3  2140.9  3417.9  7130.9 

6995.6  2180.0  3518.7  7259.8 

26 

28 

8935.4  2142.2  3421.2  7135.2 

8997.7  2181.3  3522.1  7264.1 

28 

30 

8937.5  2143.5  3424.5  7139.4 

8999.7  2182.6  3525.5  7268.5 

30 

32 

8939.6  2144  8  3427.8  7143.7 

9001.8  2183.9  3528.9  7272.8 

32 

34 

8941.6  2146.1  3431.1  7148.0 

9003.9  2185.2  3532.3  7277.2 

34 

36 

8943.7  2147.4  3434.5  7152.2 

9005.9  2186.5  3535.7  7281.5 

36 

38 

8945.8  2148.7  3437.8  7156.5 

9008.0  2187.8  3539.2  7285.9 

38 

40 

8947.9  2150.0  3441.1  7160.7 

9010.0  2189.1  3542.6  7290.3 

40 

42 

8950.0  2151.3  3444.4  7165.0 

9012.1  2190.5  3546.0  7294.6 

42 

44 

8952.1  2152.6  3447.8  7169.3 

9014.2  2191.8  3549.5  7299.0 

44 

46 

8951.1  2153.9  3451.1  7173.6 

9016.2  2193.1  3552.9  7303.4 

46 

48 

8956.2  2155.2  3454.5  7177.9 

9018.3  2194.4  3556.3  7307.7 

48 

50 

8958.3  2156.5  3457.8  7182.1 

9020.3  2195.7  3559.8  7312.1 

50 

52 

8960.4  2157.8  3461.2  7186.4 

9022.4  2197.0  3563.2  7316.5 

52 

54 

8962.5  2159.1  3464.5  7190.7 

9024.5  2198.3  3566.7  7320.9 

54 

56 

8964.5  2160.4  3467.9  7195.0 

9026.5  2199.6  3570.2  7325.3 

56 

58 

8966.6  2161.7  3471.2  7199.3 

9028.6  2200.9  3573.6  7329.7 

58 

60 

8968.7  2163.0  3474.6  7203.6 

9030.6  2202.3  3577.1  7334.1 

60 

104° 

105° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

i  _ 

0 

9030.6  2202.3  3577.1  7334.1 

9091.8  2241.8  3682.6  7467.5 

0 

2 

9032.7  2203.6  3580.5  7338.5 

9093.9  2243.1  3686.1  7472.0 

2 

4 

9034.7  2204.9  3584.0  7342.9 

9095.9  2244.4  3689.7  7476.5 

4 

6 

9036.8  2206.2  3587.5  7347.3 

9097.9  2245.8  3693.3  7481.0 

6 

8 

9038.8  2207.5  3591.0  7351.7 

9099.9  2247.1  3696.9  7485.5 

8 

10 

9040.9  2208.8  3594.4  7356.1 

9102.0  2248.4  3700.4  7490.0 

10 

12 

9042.9  2210.2  8597.9  7360.5 

9104.0  2249.7  3704.0  7494.5 

12 

14 

9045.0  2211.5  3601.4  7364.9 

9106.0  2251.1  3707.6  7499.1 

14 

16 

9047.0  2212.8  3604.9  7369.4 

9108.0  2252.4  3711.2  7503.6 

16 

18 

9049.1  2214.1  3608.4  7373.8 

9110.1  2253.7  3714.8  7508.1 

18 

20 

9051.1  2215.4  3611.9  7378.2 

9112.1  2255.0  3718.4  7512.6 

20 

22 

9053.1  2216.7  3615.4  7382.0 

9114.1  2256.4  3722.0  7517.2 

22 

24 

9055.2  2218.0  3618.9  7387.1 

9116.1  2257.7  3725.6  7521.7 

24 

26 

9057.2  2219.4  3622.4  7391.5 

9118.1  22.-9.0  3729.3  7526.3 

26 

28 

9059.3  2220.7  3625.9  7396.0 

9120.2  2260.3  3732.9  7530.8 

28 

30 

9061.3  2222.0  3629.4  7400.4 

9122.2  2261.7  3736.5  7535.3 

30 

32 

9063.3  2223.3  3G33.0  7404.8 

9124.2  2263.0  3740.1  7539  9 

32 

34 

9065.4  2224.6  3636.5  7409.3 

9126.2  2264.3  3743.7  7544.4 

34 

36 

9067.4  2226  0  3640.0  7413  8 

9128.2  2265.7  3747.4  7549.0 

36 

38 

9069.5  2227.3  3643.5  7418.2 

9130.2  2267.0  3751.0  7553.6 

88 

40 

9071.5  2228.6  3647.1  7422.7 

9132.3  2268.3  3754.6  7558.1 

40 

42 

9073.5  2229.9  3650.6  7427.1 

9134.3  2269.6  3758.3  7562.7 

42 

44 

9075.6  2231.2  3654.1  7431.6 

9136.3  2271.0  3761.9  7567.3 

44 

46 

9077.6  2232.6  3657.7  7436.1 

9138.3  2272.3  3765.6  7571.8 

46 

48 

9079.8  2233.9  3661.2  74406 

9140.3  2273.6  3769.2  7576.4 

48 

50 

9081.7  2235.2  3664.8  7445.0 

9142.3  2275.0  3772.9  7581.0 

50 

52 

9083.7  2236.5  3668.3  7449.5 

9144.3  2276.3  3776.5  7585.6 

52 

54 

9085.7  2237.8  3671.9  7454.0 

9146.3  2277.6  3780.2  7590.2 

54 

56 

9087.8  2239.2  3675.4  7458  5 

9148.3  2278.9  3783.9  7594.8 

56 

58 

9089.8  2240.5  3679.0  7463.0 

9150.4  2280.3  3787.5  7599.4 

58 

60 

9091.8  2241.8  3682.6  74675 

9152.4  2281.6  3791.2  7604.0 

60 

158  TABLE    7. —LOGARITHMS  OF  NUMBERS. 


N0123456789 


100 

1 

2 
3 

4 
5 
6 

7 
8 
9 

110 

1 


4 
5 
6 
7 
8 
9 

120 

1 
2 
3 
4 
5 
6 
7 
8 
9 

130 
1 

2 
3 
4 
5 
6 
7 
8 
9 

140 

1 
2 
3 

4 
5 
6 
7 
8 
9 

150 


00000  00043  00087  00130  00173  00217  00260  00303  00346  00389 

0432  0475  0518  0561  0604  0647  0689  0732  0775  0817 

0860  0903  0945  0988  1030  1072  1115  1157  1199  1242 

1284  1326  1368  1410  1452  1494  1536  1578  1620  1662 

1703  1745  1787  1828  1870  1912  1953  1995  2036  2078 

2119  2160  2202  2243  2284  2325  2366  2407  2449  2490 

2531  2572  2612  2653  2694  2735  2776  2816  2857  2898 

2938  2979  3019  3060  3100  3141  3181  3222  3262  3302 

3342  3383  3423  3463  3503  3543  3583  3623  3663  3703 

3743  3782  3822  3862  3902  3941  3981  4021  4060  4100 

04139  04179  04218  04258  04297  04336  04376  0441^  04454  04493 

4532  4571  4610  4650  4689  4727  4766  4805  4844  4883 

4922  4961  4999  5038  5077  5115  5154  5192  5231  5269 

6308  5346  5385  5423  6461  5500  5538  5576  5614  5652 

5690  5729  5767  5805  5843  5881  5918  5956  5994  6032 

6070  6108  6145  6183  6221  6258  6296  6333  6371  6408 

6446  6483  6521  6558  6595  6633  6670  6707  6744  6781 

6819  6856  6893  693Q  6967  7004  7041  7078  7115  7151 

7188  7225  7262  7298  7335  7372  7408  7445  7482  7518 

7555  7591  7628  7664  7700  7737  7773'  7809  7846  7882 

07918  07954  07990  08027  08063  08099  08135  08171  08207  08243 

8279  8314  8350  8386  8422  8458  8493  8529  8565  8600 

8636  8672  8707  8743  8778  8814  8849  8884  8920  8955 

8991  9026  9061  9096  9132  9167  9202  9237  9272  9307 

9342  9377  9412  9447  9482  9517  9552  9587  9621  9656 

9691  9726  9760  9795  9830  9864  9899  9934  9968  10003 

10037  10072  10106  10140  10175  10209  10243  10278  10312  0346 

0380  0415  0449  0483  0517  0551  0585  0619  0653  0687 

0721  0755  0789  0823  0857  0890  0924  0958  0992  1025 

1059  1093  1126  1160  1193  1227  1261  1294  1327  1361 

11394  11428  11461  11494  11528  11561  11594  11628  11661  11694 

1727  1760  1793  1826  1860  1893  1926  1959  1992  2024 

2057  2090  2123  2156  2189  2222  2254  2287  2320  2352 

2385  2418  2450  2483  2516  2548  2581  2613  2646  2678 

2710  2743  2775  2808  2840  2872  2905  2937  2969  3001 

3033  3066  3098  3130  3162  3194  3226  3258  3290  3322 

3354  3386  3418  3450  3481  3513  3545  3577  3609  3640 

3672  3704  3735  3767  3799  3830  3862  3893  3925  3956 

3988  4019  4051  4082  4114  4145  4176  4208  4239  4270 

4301  4333  4364  4395  4426  4457  4489  4520  4551  4582 

14613  14644  14675  14706  14737  14768  14799  14829  14860  14891 

4922  4953  4983  5014  5045  5076  5106  5137  5168  5198 

5229  5259  5290  5320  5351  5381  5412  5442  5473  5503 

5534  5564  5594  5625  5655  5685  5715  5746  5776  5806 

5836  5806  5897  5927  5957  5987  6017  6047  6077  6107 

6137  6167  6197  6227  6256  6286  6316  6346  6376  6406 

6435  6465  6495  6524  6554  6584  6613  6643  6673  6702 

6732  6761  6791  6820  6850  6879  6909  6938  6967  6997 

7026  7056  7085  7114  7143  7173  7202  7231  7260  7289 

7319  7348  7377  7406  7435  7464  7493  7522  7561  7580 

17609  17638  17667  17696  17725  17754  17782  17811  17840  17869 


TABLE  7.—  LOGARITHMS  OF  NUMBERS.  159 


N0123456789 


150 

1 
2 
3 

4 
5 
6 
7 
8 
9 


3 
4 
5 
6 
7 
8 
9 

180 

1 
2 
3 
4 
5 
6 
7 
8 
9 


17609  17638  17667  17696  17725  17754  17782  17811  17840  17869 

7898  7926  7955  7984  8013  8041  8070  8099  8127  8156 

8184  8213  8241  8270  8298  8327  8355  8384  8412  8441 

8469  8498  8526  8554  8583  8611  8639  8667  8696  8724 

8752  8780  8808  8837  8865  8893  8921  8949  8977  9005 

9033  9061  9089  9117  9145  9173  9201  9229  9257  9285 

9312  9340  9368  9396  9424  9451  9479  9507  9535  9562 

9590  9618  9645  9673  9700  9728  9756  9783  9811  9838 

9866  9893  9921  9948  99762000320030200582008520112 

2014020167201942022220249  0276  0303  0330  0358  0385 

20412  20439  20466  20493  20520  20548  20575  20602  20629  20656 

0683  0710  0737  0763  0790  0817  0844  0871  0898  0925 

0952  0978  1005.1032  1059  1085  1112  1139  1165  1192 

1219  1245  1272  1299  1325  1352  1378  1405  1431  1458 

1484  1511  1537  1564  1590  1617  1643  1669  1696  1722 

1748  1775  1801  1827  1854  1880  1906  1932  1958  1985 

2011  2037  2063  2089  2115  2141  2167  2194  2220  2246 

2272  2298  2324  2350  2376  2401  2427  2453  2479  2505 

2531  2557  2583  2608  2634  2660  2686  2712  2737  2763 

2789  2814  2840  2866  2891  2917  2943  -2968  2994  3019 

23045  23070  23096  23121  23147  23172  23198  23223  23249  23274 

3300  3325  3350  3376  3401  3426  3452  3477  3502  3528 

3553  3578  3603  3629  3654  3679  3704  3729  3754  3779 

3805  3830  3855  3880  3905  3930  3955  3980  4005  4030 

4055  4080  4105  4130  4155  4180  4204  4229  4254  4279 

4304  4329  4353  4378  4403  4428  4452  4477  4502  4527 

4551  4576  4601  4625  4650  4674  4699  4724  4748  4773 

4797  4822  4846  4871  4895  4920  4944  4969  4993  5018 

5042  5066  5091  5115  5139  5164  5188  5212  5237  5261 

5285  5310  5334  6358  5382  5406  5431  5455  5479  6503 

25527  25551  25575  25600  25624  25648  25672  25696  25720  25744 

5768  6792  5816  6840  5864  5888  6912  5935  5959  5983 

6007  6031  6055  6079  6102  6126  6150  6174  6198  6221 

6245  6269  6293  6316  6340  6364  6387  6411  6435  6458 

6482  6505  6529  6553  6576  6600  6623  6647  6670  6694 

6717  6741  6764  6788  6811  6834  6858  6881  6905 

6951  6975  6998  7021  7045  7068  7091  7114  7138  7161 

7184  7207  7231  7254  7277  7300  7323  7346  7370  7393 

7416  7439  7462  7485  7508  7531  7554  7577  7600  7623 

7646  7669  7692  7715  7738  7761  7784  7807  7830  7852 

27875  27898  27921  27944  27967  27989  28012  28035  28058  28081 

8103  8126  8149  8171  8194  8217  8240  8262  8285  8307 

8330  8353  8375  8398  8421  8443  8466  8488  8511  8533 

8556  8578  8601  8623  8646  8668  8691  8713  8735  8758 

8780  8803  8825  8847  8870  8892  8914  8937  8959  8981 

9003  9026  9048  9070  9092  9115  9137  9159  9181  9203 

9226  9248  9270  9292  9314  9336  9358  9380  9403  9425 

9447  9469  9491  9513  9535  9557  9579  9601  9623  9645 

9667  9688  9710  9732  9754  9776  9798  9820  9842  9863 

9885  9907  9929  9951  9973  999430016300383006030081 

30103  30125  30146  30168  30190  30211  30233  30255  30276  30298 


160  TABLE  7. —LOGARITHMS  OF  NUMBERS. 


N0123456789 


200 

1 
2 
3 
4 
5 
6 
7 
8 
9 

210 

1 
2 
3 
4 
5 
6 
7 
8 
9 

220 

1 
2 
3 
4 
5 
6 
7 
8 
9 

280 

1 
2 
3 
4 
5 
6 
7 
8 
9 

240 

1 
2 
3 
4 
5 
6 
7 


30103  30125  30146  30168  30190  30211  30233  30255  30276  30298 

0320  0341  0363  0384  0406  0428  0449  0471  0492  0514 

0535  0557  0578  0600  0621  0643  0664  0685  0707  0728 

0750  0771  0792  0814  0835  0856  0878  0899  0920  0942 

0963  0984  1006  1027  1048  1069  1091  1112  1133  1154 

1175  1197  1218  1239  1260  1281  1302  1323  1345  1366 

1387  1408  1429  1450  1471  1492  1513  1534  1555  1576 

1597  1618  1639  1660  1681  1702  1723  1744  1765  1785 

1806  1827  1848  1869  1890  1911  1931  1952  1973  1994 

2015  2035  2056  2077  2098  2118  2139  2160  2181  2201 

32222  32243  32263  32284  32305  32325  32346  32366  32387  32408 

2428  2449  2469  2490  2510  2531  2552  2572  2593  2613 

2634  2654  2675  2695  2715  2736  2756  2777  2797  2818 

2838  2858  2879  2899  2919  2940  2960  2980  3001  3021 

3041  3062  3082  3102  3122  3143  3163  3183  3203  3224 

3244  3264  3284  3304  3325  3345  3365  3385  3405  3425 

3445  3465  3486  3506  3526  3546  3566  3586  3606  3626 

3646  3666  3686  3706  3726  3746  3766  3786  3806  3826 

3846  3866  3885  3905  3925  3945  3965  3985  4005  4025 

4044  4064  4084  4104  4124  4143  4163  4183  4203  4223 

34242  34262  34282  34301  34321  34341  34361  34380  34400  34420 

4439  4459  4479  4498  4518  4537  4557  4577  4596  4616 

4635  4655  4674  4694  4713  4733  4753  4772  4792  4811 

4830  4850  4869  4889  4908  4928  4947  4967  4986  5005 

5025  5044  5064  5083  5102  6122  5141  5160  5180  5199 

5218  5238  5257  5276  5295  5315  5334  5353  5372  5392 

5411  5430  5449  5468  5488  5507  5526  5545  5564  5583 

5603  5622  5641  5660  5679  5698  5717  5736  5755  5774 

5793  5813  5832  5851  5870  5889  5908  5927  5946  5965 

5984  6003  6021  6040  6059  6078  6097  6116  6135  6154 

36173  36192  36211  36229  36248  36267  36286  36305  36324  36342 

6361  6380  6399  6418  6436  6455  6474  6493  6511  6530 

6549  6568  6586  6605  6624  6642  6661  6680  6698  6717 

6736  6754  6773  6791  6810  6829  6847  6866  6884  6903 

6922  6940  6959  6977  6996  7014  7033  7051  7070  7088 

7107  7125  7144  7162  7181  7199  7218  7236  7254  7273 

7291  7310  7328  7346  7365  7383  7401  7420  7438  7457 

7475  7493  7511  7530  7548  7566  7585  7603  7621  7639 

7658  7676  7694  7712  7731  7749  7767  7785  7803  7822 

7840  7858  7876  7894  7912  7931  7949  7967  7985  8003 

38021  38039  38057  38075  38093  38112  38130  38148  38166  38184 

8202  8220  8238  8256  8274  8292  8310  8328  8346  8364 

8382  8399  8417  8435  8453  8471  8489  8507  8525  8543 

8561  8578  8596  8614  8632  8650  8668  8686  8703  8721 

8739  8757  8775  8792  8810  8828  8846  8863  8881  8899 

8917  8934  8952  8970  8987  9005  9023  9041  9058  9076 

9094  9111  9129  9146  9164  9182  9199  9217  9235  9252 

9270  9287  9305  9322  9340  9358  9375  9393  9410  9428 


9445  9463  9480  9498  9515  9533  9550  9568  9585  9602 
9620  9637  9655  9672  9690  9707  9724  9742  9759  9777 

250  39794  39811  39829  39846  39863  39881  39898  39915  39933  39950 


TABLE  7.— LOGARITHMS  OF  NUMBERS.  161 


O12345678 


250 
1 

2 
3 

4 
5 
6 
7 
8 
9 

260 
1 

2 
3 

4 
5 
6 
7 
8 
9 


3 
4 
5 
6 
7 
8 
9 

280 

1 
2 
3 
4 
5 
6 
7 
8 
9 


39794  39811  39829  39846  39863  39881  39898  39915  39933  39950 

9967  9985  40002  40019  40037  40054  40071  40088  40106  40123 

4014040157  0175  0192  0209  0226  0243  0261  0278  0295 

0312  0329  0346  0364  0381  0398  0415  0432  0449  0466 

0483  0500  0518  0535  0552-  0569  0586  0603  0620  0637 

0654  0671  0688  0705  0722  0739  0756  0773  0790  0807 

0824  0841  0858  0875  0892  0909  0926  0943  0960  0976 

0993  1010  1027  1044  1061  1078  1095  1111  1128  1145 

1162  1179  1196  1212  1229  1246  1263  1280  1296  1313 

1330  1347  1363  1380  1397  1414  1430  1447  1464  1481 

41497  41514  41531  41547  41564  41581  41597  41614  41631  41647 

1664  1681  1697  1714  1731  1747  1764  1780  1797  1814 

1830  1847  1863  1880  1896  1913  1929  1946  1963  1979 

1996  2012  2029  2045  2062  2078  2095  2111  2127  2144 

2160  2177  2193  2210  2226  2243  2259  2275  2292  2308 

2325  2341  2357  2374  2390  2406  2423  2439  2455  2472 

2488  2504  2521  2537  2553  2570  2586  2602  2619  263£ 

2651  2667  -2684  2700  2716  2732  2749  2765  2781  2797 

2813  2830  2846  2862  2878  2894  2911  2927  2943  2959 

2975  2991  3008  3024  3040  3056  3072  3088  3104  3120 

43136  43152  43169  43185  43201  43217  43233  43249  43265  43281 

3297  3313  3329  3345  3361  3377  3393  3409  3425  3441 

3457  3473  3489  3505  3521  3537  3553  3569  3584  3600 

3616  3632  3648  3664  3680  3696  3712  3727  3743  3759 

3775  3791  3807  3823  3838  3854  3870  3886  3902  3917 

3933  3949  3965  3981  3996  4012  4028  4044  4059  4075 

4091  4107  4122  4138  4154  4170  4185  4201  4217  4232 

4248  4264  4279  4295  4311  4326  4342  4358  4373  4389 

4404  4420  4436  4451  4467  4483  4498  4514  4529  4545 

4560  4576  4592  4607  4623  4638  4654  4669  4685  4700 

44716  44731  44747  44762  44778  44793  44809  44824  44840  44855 

4871  4886  4902  4917  4932  4948  4963  4979  4994  5010 

5025  5040  5056  5071  5086  5102  5117  5133  5148  5163 

5179  5194  5209  5225'  5240  5255  5271  5286  5301  5317 

5332  5347  5362  5378  5393  5408  5423  5439  5454  5469 

5484  5500  5515  5530  5545  5561  5576  5591  5606  5621 

5637  5652  5667  5682  5697  5712  5728  5743  5758  5773 

5788  5803  5818  5834  5849  5864  5879  5894  5909  5924 

5939  5954  5969  5984  6000  6015  6030  6045  6060  6075 

6090  6105  6120  6135  6150  6165  6180  6195  6210  6225 

46240  46255  46270  46285  46300  46315  46330  46345  46359  46374 

6389  6404  6419  6434  6449  6464  6479  6494  6509  6523 

6538  6553  6568  6583  6598  6613  6627  6642  6657  6672 

6687  6702  6716  6731  6746  6761  6776  6790  6805  6820 

6835  6850  6864  6879  6894  6909  6923  6938  6953  6967 

6982  6997  7012  7026  7041  7056  7070  7085  7100  7114 

7129  7144  7159  7173  7188  7202  7217  7232  7246  7261 

7276  7290  7305  7319  7334  7349  7363  7378  7392  7407 

7422  7436  7451  7465  7480  7494  7509  7524  7538  7553 

7567  7582  7596  7611  7625  7640  7654  7669  7683  7698 

47712  47727  47741  47756  47770  47784  47799  47813  47828  47842 


162  TABLE  7.— LOGARITHMS   OF   NUMBERS. 


O12345678 


300 
1 
2 
3 

4 
5 
6 

7 
8 
9 

310 
1 

2 
3 
4 
5 
6 
7 
8 
9 

320 

1 
2 
3 
4 
5 
6 
7 
0 
9 

330 

1 
2 
3 
4 
5 
G 
7 
8 
9 

340 

1 

2 
3 

4 
5 
6 
7 
8 
9 

350 


47712  47727  47741  47756  47770  47784  47799  47813  47828  47842 

7857  7871  7885  7900  7914  7929  7943  7958  7972  7986 

8001  8015  8029  8044  8058  8073  8087  8101  8116  8130 

8144  8159  8173  8187  8202  8216  8230  8244  8259  8273 

8287  8302  8316  8330-  8344  8359  8373  8387  8401  8416 

8430  8444  8458  8473  8487  8501  8515  8530  8544  8558 

8572  8586  8601  8615  8629  8643  8657  8671  8686  87QO 

8714  8728  8742  8756  8770  8785  8799  8813  8827  8841 

8855  8869  8883  8897  8911  8926  8940  8954  8968  8982 

8996  9010  9024  9038  9052  9066  9080  9094  9108  9122 

49136  49150  49164  49178  49192  49206  49220  49234  49248  49262 

9276  9290  9304  9318  9332  9346  9360  9374  9388  9402 

9415  9429  9443  9457  9471  9485  9499  9513  9527  9541 

9554  9568  9582  9596  9610  9624  9638  9651  9665  9679 

9693  9707  9721  9734  9748  9762  9776  9790  9803  9817 

9831  9845  9859  9872  9886  9900  9914  9927  9941  9955 

9969  9982  9996  50010  50024  50037  50051  50065  50079  50092 

501065012050133  0147  0161  0174  0188  0202  0215  0229 

0243  0256  0270  0284  0297  0311  0325  0338  0352  0365 

0379  0393  0406  0420  0433  0447  0461  0474  0488  0501 

50515  50529  50542  50556  50569  50583  50596  50610  50623  50637 

0651  0664  0678  0691  0705  0718  0732  0745  0759  0772 

0786  0799  0813  0826  0840  0853  0866  0880  0893  0907 

0920  0934  0947  0961  0974  0987  1001  1014  1028  1041 

1055  1068  1081  1095  1108  1121  1135  1148  1162  1175 

1188  1202  1215  1228  1242  1255  1268  1282  1295  1308 

1322  1335  1348  1362  1375  1388  1402  1415  1428  1441 

1455  1468  1481  1495  1508  1521  1534  1548  1561  1574 

1587  1601  1614  1627  1640  1654  1667  1680  1693  1706 

1720  1733  1746  1759  1772  1786  1799  1812  1825  1838 

51851  51865  51878  51891  51904  51917  51930  51943  51957  51970 

1983  1996  2009  2022  2035  2048  2061  2075  2088  2101 

2114  2127  2140  2153  2166  2179  2192  2205  2218  2231 

2244  2257  2270  2284  2297'  2310  2323  2336  2349  2362 

2375  2388  2401  2414  2427  2440  2453  2466  2479  2492 

2504  2517  2530  2543  2556  2569  2582  2595  2608  2621 

2634  2647  2660  2673  2686  2699  2711  2724  2737  2750 

2763  2776  2789  2802  2815  2827  2840  2853  2866  2879 

2892  2905  2917  2930  2943  2956  2969  2982  2994  3007 

3020  3033  3046  3058  3071  3084  3097  3110  3122  3135 

53148  53161  53173  53186  53199  53212  53224  53237  53250  53263 

3275  3288  3301  3314  3326  3339  3352  3364  3377  3390 

3403  3415  3428  3441  3453  3466  3479  3491  3504  3517 

3529  3542  3555  3567  3580  3593  3606  3618  3631  3643 

3656  3668  3681  3694  3706  3719  3732  3744  3757  3769 

3782  3794  3807  3820  3832  3845  3857  3870  3882  3895 

3908  3920  3933  3945  3958  3970  3983  3995  4008  4020 

4033  4045  4058  4070  4083  4095  4108  4120  4133  4145 

4158  4170  4183  4195  4208  4220  4233  4245  4258  4270 

4283  4295  4307  4320  4832  4345  4357  4370  4382  4394 

54407  54419  54432  54444  54456  54469  54481  54494  54506  54518 


TABLE  7.— LOGARITHMS  OF  NUMBERS.  163 


N    0123456789 


350 
1 

2 
3 
4 
5 
6 
7 
8 
9 

360 

1 
2 
3 

4 
5 
6 

7 
8 
9 

370 

1 
2 

3 

4 
5 
6 
7 
8 
9 

380 

1 
2 
3 

4 
5 
6 
7 
8 
9 

390 

1 
2 
3 
4 
5 
6 
7 
8 
9 


54407  54419  54432  54444  54456  54469  54481  54494  54506  54518 

4531  4543  4555  4568  4580  4593  4605  4617  4630  4642 

4654  4667  4679  4691  4704  4716  4728  4741  4753  4765 

4777  4790  4802  4814  4827  4839  4851  4864  4876  4888 

4900  4913  4925  4937  4949  4962  4974  4986  4998  6011 

5023  5035  5047  5060  5072  5084  5096  5108  5121  5133 

5145  5157  5169  5182  5194  5206  5218  5230  5242  5255 

5267  5279  5291  5303  5315  5328  6340  5352  6364  5376 

5388  5400  5413  5425  5437  6449  5461  5473  5485  5497 

5509  5522  5534  5546  5558  6570  5582  6594  5606  6618 

55630  55642  55654  55666  55678  65691  55703  55715  55727  55739 

5751  5763  5775  6787*  5799  5811  5823  5835  5847  5859 

5871  5883  5895  5907  5919  5931  5943  5955  5967  5979 

5991  6003  6015  6027  6038  6050  6062  6074  6086  6098 

6110  6122  6134  6146  6158  6170  6182  6194  6205  6217 

6229  6241  .  6253  6265  6277  6289  6301  6312  6324  6336 

6348  6360  6372  6384  6396  6407  6419  6431  6443  6455 

6467  6478  6490  6502  6514  6526  6538  6549  6561  6573 

6585  6597  6608  6620  6632  6644  6656  6667  6679  6691 

6703  6714  6726  6738  6750  6761  6773  6785  6797  6808 

56820  56832  56844  56855  56867  56879  56891  56902  56914  56926 

6937  6949  6961  6972  6984  6996  7008  7019  7031  7043 

7054  7066  7078  7089  7101  7113  7124  7136  7148  7159 

7171  7183  7194  7206  7217  7229  7241  7252  7264  7276 

7287  7299  7310  7322  7334  7345  7357  7368  7380  7392 

7403  7415  7426  7438  7449  7461  7473  7484  7496  7507 

7519  7530  7542  7553  7565  7576  7588  7600  7611  7623 

7634  7646  7657  7669  7680  7692  7703  7715  7726  7738 

7749  7761  7772  7784  7795  7807  7818  7830  7841  7852 

7864  7875  7887  7898  7910  7921  7933  7944  7955  7967 

57978  57990  58001  58013  58024  58035  58047  58058  58070  58081 

8092  8104  8115  8127  8138  8149  8161  8172  8184  8195 

8206  8218  8229  8240  8252  8263  8274  8286  8297  8309 

8320  8331  8343  8354  8365  8377  8388  8399  8410  8422 

8433  8444  8456  8467  8478  8490  8501  8512  8524  8535 

8546  8557  8569  8580  8591  8602  8614  8625  8636  8647 

8659  8670  8681  8692  8704  8715  8726  8737  8749  8760 

8771  8782  8794  8805  8816  8827  8838  8850  8861  8872 

8883  8894  8906  8917  8928  8939  8950  8961  8973  8984 

8995  9006  9017  9028  9040  9051  9062  9073  9084  9095 

59106  59118  59129  59140  59161  59162  59173  59184  59195  59207 

9218  9229  9240  9251  9262  9273  9284  9295  9306  9318 

9329  9340  9351  9362  9373  9384  9395  9406  9417  9428 

9439  9450  9461  9472  9483  9494  9506  9517  9528  9539 

9550  9561  9572  9583  9594  9605  9616  9627  9638  9649 

9660  9671  9682  9693  9704  9715  9726  9737  9748  9759 

9770  9780  9791  9802  9813  9824  9835  9846  9857  9868 

9879  9890  9901  9912  9923  9934  9945  9956  9966  9977 

9988  9999  60010  60021  60032  60043  60054  60065  60076  60086 

60097  60108  0119  0130  0141  0152  0163  0173  0184  0195 


400  60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 


164  TABLE  7. —LOGARITHMS  OF  NUMBERS. 


NO123456789 


400 

1 
2 
3 

4 
5 
6 
7 
8 
9 

410 

1 
2 
3 

4 
5 
6 
7 
8 
9 

420 

1 
2 
3 

4 
5 
6 
7 
8 
9 

430 
1 
2 
3 

4 
5 
6 
7 
8 
9 

440 
1 

2 
3 

4 
5 
6 
7 
8 
9 

450 


60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 

0314  0325  0336  0347  0358  0369  0379  0390  0401  0412 

0423  0433  0444  0455  0466  0477  0487  0498  0509  0520 

0531  0541  0552  0563  0574  0584  0595  0606  0617  0627 

0638  0649  0660  0670  0681  0692  0703  0713  0724  0735 

0746  0756  0767  0778  0788  0799  0810  0821  0831  0842 

0853  0863  0874  0885  0895  0906  0917  0927  0938  0949 

0959  0970  0981  0991  1002  1013  1023  1034  1045  1055 

1066  1077  1087  1098  1109  1119  1130  1140  1151  1162 

1172  1183  1194  1204  1215  1225  1236  1247  1257  1268 

61278  61289  61300  61310  61321  61331  61342  61352  61363  61374 

1384  1395  1405  1416  1426  1437  1448  1458  1469  1479 

1490  1500  1511  1521  1532  1542  1553  1563  1574  1584 

1595  1606  1616  1627  1637  1648  1658  1669  1679  1690 

1700  1711  1721  1731  1742  1752  1763  1773  1784  1794 

1805  1815  1826  1836  1847  1857  1868  1878  1888  1899 

1909  1920  1930  1941  1951  1962  1972  1982  1993  2003 

2014  2024  2034  2045  2055  2066  2076  2086  2097  2107 

2118  2128  2138  2149  2159  2170  2180  2190  2201  2211 

2221  2232  2242  2252  2263  2273  2284  2294  2304 

62325  62335  62346  62356  62366  62377  62387  62397  62408  62418 

2428  2439  2449  2459  2469  2480  2490  2500  2511  2521 

2531  2542  2552  2562  2572  2583  2593  2603  2613  2624 

2634  2644  2655  2665  2675  2685  2696  2706  2716  2726 

2737  2747  2757  2767  2778  2788  2798  2808  2818  2829 

2839  2849  2859  2870  2880  2890  2900  2910  2921  2931 

2941  2951  2961  2972  2982  2992  3002  3012  3022  3033 

3043  3053  3063  3073  3083  3094  3104  3114  3124  3134 

3144  3155  3165  3175  3185  3195  3205  3215  3225  3236 

3246  3256  3266  3276  3286  3296  3306  3317  3327  3337 

63347  63357  63367  63377  63387  63397  63407  63417  63428  63438 

3448  3458  3468  3478  3488  3498  3508  3518  3528  3538 

3548  3558  3568  3579  3589  3599  3609  3619  3629  3639 

3649  3659  3669  3679  3689  3699  3709  3719  3729  3739 

3749  3759  3769  3779  3789  3799  3809  3819  3829  3839 

3849  3859  3869  3879  3889  3899  3909  3919  3929  3939 

3949  3959  3969  3979  3988  3998  4008  4018  4028  4038 

4048  4058  4068  4078  4088  4098  4108  4118  4128  4137 

4147  4157  4167  4177  4187  4197  4207  4217  4227  4237 

4246  4256  4266  4276  4286  4296  4306  4316  4326  4335 

64345  64355  64365  64375  64385  64395  64404  64414  64424  64434 

4444  4454  4464  4473  4483  4493  4503  4513  4523  4532 

4542  4552  4562  4572  4582  4591  4601  4611  4621  4631 

4640  4650  4660  4fc70  4680  4689  4699  4709  4719  4729 

4738  4748  4758  4768  4777  4787  4797  4807  4816  4826 

4836  4846  4856  4865  4875  4885  4895  4904  4914  4924 

4933  4943  4953  4963  4972  4982  4992  5002  5011  5021 

5031  5040  5050  5060  5070  5079  5089  5099  5108  5118 

5128  6137  5147  5157  5167  5176  5186  5196  5205  5215 

5225  6234  5244  5254  5263  5273  5283  5292  5302  5312 

65321  65331  65341  65350  65360  65369  65379  65389  65398  65408 


TABLE  7. -LOGARITHMS  OF  NUMBERS.  165 


0123456789 


450 

1 
2 
3 

4 
5 
6 

7 
8 
9 

460 
1 
2 
3 
4 
5 
6 
7 
% 
9 


3 

4 
5 
6 
7 
8 
9 

500 


65321  65331  65341  65350  65360  65369  65379  65389  65398  65408 

5418  5427  5437  5447  5456  5466  5475  5485  5495  5504 

5514  5523  5533  5543  5552  5562  5571  5581  5591  5600 

5610  5619  5629  5639  5648  5658  5667  5677  5686  5696 

5706  5715  5725  5734  5744  5753  5763  6772  5782  5792 

5801  5811  5820  5830  5839  5849  5858  5868  6877  6887 

5896  5906  5916  5925  5935  5944  5954  5963  5973  5982 

5992  6001  6011  6020  6030  6039  6049  6058  6068  6077 

6087  6096  6106  6115  6124  6134  6143  6153  6162  6172 

6181  6191  6200  6210  6219  6229  6238  6247  6257  6266 

66276  66285  66295  66304  66314  66323  66332  66342  66351  66361 

6370  6380  6389  6398  T6408  6417  6427  6436  6445  6455 

6464  6474  6483  6492  6502  6511  6521  6530  6539  6549 

6558  6567  6577  6586  6596  6605  6614  6624  6633  6642 

6652  6661  6671  6680  6689  6699  6708  6717  6727  6736 

6745  6755  6764  6773  6783  6792  6801  6811  6820  6829 

6839  6848  6857  6867  6876  6885  6894  6904  6913  6922 

6932  6941  6950  6960  6969  6978  6987  6997  7006  7015 

7025  7034  7043  7052  7062  7071  7080  7089  7099  7108 

7117  7127  7136  7145  7154  7164  7173  7182  7191  7201 

67210  67219  67228  67237  67247  67256  67265  67274  67284  67293 

7302  7311  7321  7330  7339  7348  7357  7367  7376  7385 

7394  7403  7413  7422  7431  7440  7449  7459  7468  7477 

7486  7495  7504  7514  7523  7532  7541  7550  7560  7569 

7578  7587  7596  7605  7614  7624  7633  7642  7651  7660 

7669  7679  7688  7697  7706  7715  7724  7733  7742  7752 

7761  7770  7779  7788  7797  7806  7815  7825  7834  7843 

7852  7861  7870  7879  7888  7897  7906  7916  7925  7934 

7943  7952  7961  7970  7979  7988  7997  8006  8015  8024 

8034  8043  8052  8061  8070  8079  8088  8097  8106  8115 

68124  68133  68142  68151  68160  68169  68178  68187  68196  68205 

8215  8224  8233  8242  8251  8260  8269  8278  8287  8296 

8305  8314  8323  8332  8341  8350  8359  8368  8377  8386 

8395  8404  8413  8422  8431  8440  8449  8458  8467  8476 

8485  8494  8502  8511  8520  8529  8538  8547  8556  8565 

8574  8583  8592  8601  8610  8619  8628  8637  8646  8655 

8664  8673  8681  8690  8699  8708  8717  8726  8735  8744 

8753  8762  8771  8780  8789  8797  8806  8815  8824  8833 

8842  8851  8860  8869  8878  8886  8895  8904  8913  8922 

8931  8940  8949  8958  8966  8975  8984  8993  9002  9011 

69020  69028  69037  69046  69055  69064  69073  69082  69090  69099 

9108  9117  9126  9135  9144  9152  9161  9170  9179  9188 

9197  9205  9214  9223  9232  9241  9249  9258  9267  9276 

9285  9294  9302  9311  9320  9329  9338  9346  9355  9364 

9373  9381  9390  9399  9408  9417  9425  9434  9443  $452 

9461  9469  9478  9487  9496  9504  9513  9522  9531  9539 

9548  9557  9566  9574  9583  9592  9601  9609  9618  9627 

9636  9644  9653  9662  9671-  9679  9688  9697  9705  9714 

9723  9732  9740  9749  9758  9767  9775  9784  9793  9801 

9810  9819  9827  9836  9845  9854  9862  9871  9880 

69897  69906  69914  69923  69932  69940  69949  69958  69966  69975 


166  TABLE  7.— LOGARITHMS   OF  NUMBERS. 


0123456789 


500 

1 
2 
3 
4 
5 
6 
7 
8 
9 

510 

1 
2 
3 

4 
5 
6 
7 
8 
9 

520 

1 
2 
3 

4 
5 
6 
7 
8 
9 

530 

1 

2 
3 
4 
5 
6 
7 
8 
9 

540 

1 

2 
3 

4 
5 
G 
7 
8 
9 

550 


69897  69906  69914  69923  69932  69940  69949  69958  69966  69975 

9984  9992  70001  70010  70018  70027  70036  70044  70053  70062 

70070  70079  0088  0096  0105  0114  0122  0131  0140  0148 

0157  0165  0174  0183  0191  0200  0209  0217  0226  0234 

0243  0252  0260  0269  0278  0286  0295  0303  0312  0321 

0329  0338  0346  0355  0364  0372  0381  0389  0398  0406 

0415  0424  0432  0441  0449  0458  0467  0475  0484  0492 

0501  0509  0518  0526  0535  0544  0552  0561  0569  0578 

0586  0595  0603  0612  0621  0629  0638  0646  065£  0663 

0672  0680  0689  0697  0706  0714  0723  0731  0740  0749 

70757  70766  70774  70783  70791  70800  70808  70817  70825  70834 

0842  0851  0859  0868  0876  0885  0893  0902  0910  0919 

0927  0935  0944  0952  0961  0969  0978  0986  0995  1003 

1012  1020  1029  1037  1046  1054  1063  1071  1079  1088 

1096  1105  1113  1122  1130  1139  1147  1155  1164  1172 

1181  1189  1198  1206  1214  1223  1231  1240  1248  1257 

1273  1282  1290  1299  1307  1315  1324  1332  1341 

1349  1357  1366  1374  1383  1391  1399  1408  1416  1425 

1433  1441  1450  1458  1466  1475  1483  1492  1500  1508 

1517  1525  1533  1542  1550  1559  1567  1575  1584  1592 

71600  71609  71617  71625  71634  71642  71650  71659  71667  71675 

1684  1692  1700  1709  1717  1725  1734  1742  1750  1759 

1767  1775  1784  1792  1800  1809  1817  1825  1834  1842 

1850  1858  1867  1875  1883  1892  1900  1908  1917  1925 

1933  1941  1950  1958  1966  1975  1983  1991  1999  2008 

2016  2024  2032  2041  2049  2057  2066  2074  2082  2090 

2099  2107  2115  2123  2132  2140  2148  2156  2165  2173 

2181  2189  2198  2206  2214  2222  2230  2239  2247  2255 

2263  2272  2280  2288  2296  2304  2313  2321  2329  2337 

2346  2354  2362  2370  2378  2387  2395  2403  2411  2419 

72428  72436  72444  72452  72460  72469  72477  72485  72493  72501 

2509  2518  2526  2534  2542  2550  2558  2567  2575  2583 

2591  2599  2607  2616  2624  2632  2640  2648  2656  2665 

2673  2681  2689  2697  2705  2713  2722  2730  2738  2746 

2754  2762  2770  2779  2787  2795  2803  2811  2819  2827 

2835  2843  2852  2860  2868  2876  2884  2892  2900  2908 

2916  2925  2933  2941  2949  2957  2965  2973  2981  2989 

2997  3006  3014  3022  3030  3038  3046  3054  3062  3070 

3078  3086  3094  3102  3111  3119  3127  313^  3143  3151 

3159  3167  3175  3183  3191  3199  3207  3215  3223  3231 

73239  73247  73255  73263  73272  73280  73288  73296  73304  73312 

3320  3328  3336  3344  3352  3360  3368  3376  3384  3392 

3400  3408  3416  3424  3432  3440  3448  3456  3464  3472 

3480  3488  3496  3504  3512  3520  3528  3536  3544  3552 

3560  3568  3576  3584  3592  3600  3608  3616  3624  3632 

3640  3648  3656  3664  3672  3679  3687  3695  3703  3711 

3719  3727  3735  3743  3751  3759  3767  377£  3783  3791 

3799  3807  3815  3823  3830  3838  3846  3854  3862  3870 

3878  3886  3894  3902  3910  3918  3926  3933  3941  3949 

3957  3965  3973  3981  3989  3997  400£  4013  4020  4028 

74036  74044  74052  74060  74068  74076  74084  74092  74099  74107 


TABLE  7.— LOGARITHMS  OF  NUMBERS  167 


NO1234:56789 


550 

1 
2 

3 

4 
5 
6 
7 
8 
9 

560 

1 
2 
3 
4 
5 
6 
7 
8 
9 

570 

1 
2 
3 
4 
5 
6 
7 
8 
9 

580 

1 
2 
3 

4 
5 
6 
7 
8 
9 

590 

1 
2 
3 
4 
5 
6 
7 
8 
9 


74036  74044  74052  74060  74068  74076  74084  74092  74099  74107 

4115  4123  4131  4139  4147  4155  4162  4170  4178  4186 

4194  4202  4210  4218  4225  4233  4241  4249  4257  4265 

4273  4280  4288  4296  4304  4312  4320  4327  4335  4343 

4351  4359  4367  4374  4382  4390  4398  4406  4414  4421 

4429  4437  4445  4453  4461  4468  4476  4484  4492  4500 

4507  4515  4523  4531  4539  4547  4554  4562  4570  4578 

4586  4593  4601  4609  4617  4624  4632  4640  4648  4656 

4663  4671  4679  4687  4695  4702  4710  4718  4726  4733 

4741  4749  4757  4764  4772  4780  4788  4796  4803  4811 

74819  74827  74834  74842  74$50  74858  74865  74873  74881  74889 

4896  4904  4912  4920  4927  4935  4943  4950  4958  4966 

4974  4981  4989  4997  5005  5012  5020  5028  6035  5043 

5051  5059  5066  5074  5082  5089  5097  5105  5113  5120 

5128  5136  5143  5151  5159  5166  5174  5182  5189  5197 

5205  5213  5220  5228  5236  5243  5251  5259  5266  5274 

5282  5289  5297  5305  5312  5320  5328  5335  5343  5351 

5358  5366  5374  5381  5389  5397  5404  5412  6420  5427 

5435  5442  5450  5458  5465  5473  5481  5488  5496  5504 

5511  5519  5526  5534  5542  5549  5557  556$  5572  5580 

75587  75595  75603  75610  75618  75626  75633  75641  75648  75656 

5664  5671  5679  5686  5694  5702  5709  5717  5724  5732 

5740  5747  5755  5762  5770  5778  5785  5793  5800  5808 

5815  5823  5831  5838  5846  5853  5861  5868  5876  5884 

5891  5899  5906  5914  5921  5929  5937  5944  5952  5959 

5967  5974  5982  5989  5997  6005  6012  6020  6027  6035 

6042  6050  6057  6065  6072  6080  6087  6095  6103  6110 

6118  6125  6133  6140  6148  6155  6163  6170  6178  6185 

6193  6200  6208  6215  6223  6230  6238  6245  6253  6260 

6268  6275  6283  6290  6298  6305  6313  6320  6328  6335 

76343  76350  76358  76365  76373  76380  76388  76395  76403  76410 

6418  6425  6433  6440  6448  6455  6462  6470  6477  6485 

6492  6500  6507  6515  6522  6530  6537  6545  6552  6559 

6567  6574  6582  6589  6597  6604  6612  6619  6626  6634 

6641  6649  6656  6664  6671  6678  6686  6693  6701  6708 

6716  6723  6730  6738  6745  6753  6760  6768  677$  6782 

6790  6797  6805  6812  6819  6827  6834  6842  6849  6856 

6864  6871  6879  6886  6893  6901  6908  6916  6923  6930 

6938  6945  6953  6960  6967  6975  6982  6989  6997  7004 

7012  7019  7026  7034  7041  7048  7056  7063  7070  7078 

77085  77093  77100  77107  77115  77122  77129  77137  77144  77151 

7159  7166  7173  7181  7188  7195  7203  7210  7217  722$ 

7232  7240  7247  7254  7262  7269  7276  7283  7291  7298 

7305  7313  7320  7327  7335  7342  7349  7357  7364  7371 

7379  7386  7393  7401  7408  7415  7422  7430  7437  7444 

7452  7459  7466  7474  7481  7488  7495  7503  7510  7517 

7525  7532  7539  7546  7554  7561  7568  7576  7583  7590 

7597  7605  7612  7619  7627  7634  7641  7648  7656  7663 

7670  7677  7685  7692  7699  7706  7714  7721  7728  7735 

7743  7750  7757  7764  7772  7779  7786  7793  7801  7808 

77815  77822  77830  77837  77844  77851  77859  77866  77873  77880 


168  TABLE  7. —LOGARITHMS  OP  NUMBERS. 


O123456789 


600 
1 
2 
3 

4 
5 
6 
7 
8 
9 

610 
1 
2 
3 

4 
5 
6 

7 
8 


620 
1 
2 
3 

4 
5 
6 
7 
8 
9 


1 
2 
3 

4 
5 
6 
7 
8 
9 

640 
1 
2 
3 

4 
5 
8 
7 
8 


650 


77815  77822  77830  77837  77844  77851  77859  77866  77873  77880 

7887  7895  7902  7909  7916  7924  7931  7938  7945  7952 

7960  7967  7974  7981  7988  7996  8003  8010  8017  802$ 

8032  8039  8046  8053  8061-  8068  8075  8082  8089  8097 

8104  8111  8118  8125  8132  8140  8147  8154  8161  8168 

8176  8183  8190  8197  8204  8211  8219  8226  8233  8240 

8247  8254  8262  8269  8276  8283  8290  8297  830$  8312 

8319  8326  8333  8340  8347  8355  8362  8369  8376  8383 

8390  8398  8405  8412  8419  8426  8433  8440  8447  845$ 

8462  8469  8476  8483  8490  8497  8504  8512  8519  8526 

78533  78540  78547  78554  78561  78569  78576  78583  78590  78597 

8604  8611  8618  8625  8633  8640  8647  8654  8661  8668 

8675  8682  8689  8606  8704  8711  8718  8725  8732  8739 

8746  8753  8760  8767  8774  8781  8789  8796  8803  8810 

8817  8824  8831  8838  8845  8852  8859  8866  8873  8880 

8888  8895  8902  8909  8916  8923  8930  8937  8944  8951 

8958  8965  8972  8979  8986  8993  9000  9007  9014  9021 

9029  9036  9043  9050  9057  9064  9071  9078  9085  9092 

9099  9106  9113  9120  9127  9134  9141  9148  9155  9162 

9169  9176  9183  9190  9197  9204  9211  9218  9225  9232 

79239  79246  79253  79260  79267  79274  79281  79288  79295  79302 

9309  9316  9323  9335  9337  934  i  9351  9358  9365  9372 

9379  9386  9393  9400  9407  9414  9421  9428  9435  9442 

9449  9456  9463  9470  9477  9484  9491  9498  9505  9511 

9518  9525  9532  9539  9546  9553  9500  9567  9574  9581 

9588  9595  9602  9609  9616  9623  9630  9637  9644  9650 

9657  9664  9671  9678  9685  9692  9699  9706  9713  9720 

9727  9734  9741  9748  9754  9761  9768  9775  9782  9789 

9796  9803  9810  9817  9824  9831  9837  9844  9851  9858 

9865  9872  9879  9886  9893  9900  9906  9913  9920  9927 

79934  79941  79948  79955  79962  79969  79975  79982  79989  79996 

80003  80010  80017  80024  80030  80037  80044  80051  80058  80065 

0072  0079  0085  0092  0099  0106  0113  0120  0127  0134 

0140  0147  0154  0161  0168  0175  0182  0188  0195  0202 

0209  0216  0223  0229  0236  0243  0250  0257  0264  0271 

0277  0284  0291  0298  0305  0312  0318  0325  0332  0339 

0346  0353  0359  0366  0373  0380  0387  0393  0400  0407 

0414  0421  0428  0434  0441  0448  0455  0462  0468  0475 

0482  0489  0496  0502  0509  0516  0523  0530  0536  0543 

0550  0557  0564  0570  0577  0584  0591  0598  0604  0611 

80618  80625  80632  8063P  80645  80652  80659  80665  80672  80679 

0686  0693  0699  0706  0713  0720  0726  0733  0740  0747 

0754  0760  0767  0774  0781  0787  0794  0801  0808  0814 

0821  0828  0835  0841  0848  085$  0862  0868  0875  0882 

0889  0895  0902  0909  0916  0922  0929  0936  0943  0949 

0956  0963  0969  0976  0983  0990  0996  1003  1010  1017 

1023  1030  1037  1043  1050  1057  1064  1070  1077  1084 

1090  1097  1104  1111  1117  1124  1131  1137  1144  1151 

1158  1164  1171  1178  1184  1191  1198  1204  1211  1218 

1224  1231  1238  124$  1251  1258  126$  1271  1278  128$ 

81291  81298  8130$  81311  81318  8132$  81331  81338  8134$  81351 


TABLE  7 — LOGARITHMS  OF  NUMBERS.  169 


0123456789 


650 

1 
2 
3 
4 
5 
6 
7 
8 
9 

660 

1 
2 
3 
4 
5 
6 
7 
8 
9 


3 

4 
5 
6 
7 
8 
9 

690 

1 
2 
3 
4 
5 
6 
7 
8 
9 

700 


81291  81298  81305  81311  81318  81325  81331  81338  81345  81351 

1358  1365  1371  1378  1385  1391  1398  1405  1411  1418 

1425  1431  1438  1445  1451  1458  1465  1471  1478  1485 

1491  1498  1505  1511  1518  1525  1531  1538  1544  1551 

1558  1564  1571  1578  1584  1591  1598  1604  1611  1617 

1624  1631  1637  1644  1651  1657  1664  1671  1677  1684 

1690  1697  1704  1710  1717  1723  1730  1737  1743  1750 

1757  1763  1770  1776  1783  1790  1796  1803  1809  1816 

1823  1829  1836  1842  1849  1856  1862  1869  1875  1882 

1889  1895  1902  1908  1915  1921  1928  1935  1941  1948 

81954  81961  81968  81974  81981  81987  81994  82000  82007  82014 

2020  2027  2033  2040  2046  2053  2060  2066  2073  2079 

2086  2092  2099  2105  2112  2119  2125  2132  2138  2145 

2151  2158  2164  2171  2178  2184  2191  2197  2204  2210 

2217  2223  2230  2236  2243  2249  2256  2263  2269  2276 

2282  2289  2295  2302  2308  2315  2321  2328  2334  2341 

2347  2354  2360  2367  2373  2380  2387  2393  2400  2406 

2413  2419  2426  2432  2439  2445  2452  2458  2465  2471 

2478  2484  2491  2497  2504  2510  2517  2523  2530  2536 

2543  2549  2556  2562  2569  2575  2582  2588  2595  2601 

82607  82614  82620  82627  82633  82640  82646  82653  82659  82666 

2672  2679  2685  2692  2698  2705  2711  2718  2724  2730 

2737  2743  2750  2750  2763  2769  2776  2782  2789  2795 

2802  2808  2814  2821  2827  2834  2840  2847  2853  2860 

2866  2872  2879  2885  2892  2898  2905  2911  2918  2924 

2930  2937  2943  2950  2956  2963  2969  2975  2982  2988 

2995  3001  3008  3014  3020  3027  3033  3040  3046  3052 

3059  3065  3072  3078  3085  3091  3097  3104  3110  3117 

3123  3129  3136  3142  3149  3155  3161  3168  3174  3181 

3187  3193  3200  3206  3213  3219  3225  3232  3238  3245 

83251  83257  83264  83270  83276  83283  83289  83296  83302  83308 

3315  3321  3327  3334  3340  3347  3353  3359  3366  3372 

3378  3385  3391  3398  3404  3410  3417  3423  3429  3436 

3442  3448  3455  3461  3467  3474  3480  3487  3493  3499 

3506  3512  3518  3525  3531  3537  3544  3550  3556  3563 

3569  3575  3582  3588  3S94  3601  3607  3613  3620  3626 

3632  3639  3645  3651  3658  3664  3670  3677  3683  3689 

3696  3702  3708  3715  3721  3727  3734  3740  3746  3753 

3759  3765  3771  3778  3784  3790  3797  3803  3809  3816 

3822  3828  3835  3841  3847  3853  3860  3866  3872  3879 

83885  83891  83897  83904  83910  83916  83923  83929  83935  83942 

3948  3954  3960  3967  3973  3979  3985  3992  3998  4004 

4011  4017  4023  4029  4036  4042  4048  4055  4061  4067 

4073  4080  4086  4092  4098  4105  4111  4117  4123  4130 

4136  4142  4148  4155  4161  4167  4173  4180  4186  4192 

4198  4205  4211  4217  4223  4230  4236  4242  4248  4255 

4261  4267  4273  4280  4286  4292  4298  4305  4311  4317 

4323  4330  4336  4342  4348  4354  4361  4367  4373  4379 

4386  4392  4398  4404  4410  4417  4423  4429  4435  4442 

4448  4454  4460  4466  4473  4479  448£  4491  4497  4504 

84510  84516  84522  84528  84535  84541  84547  84553  84559  84566 


170  TABLE  7.— LOGARITHMS  OF  NUMBERS. 


N0123456789 


700 
1 
2 
3 

4 
5 
6 

7 
8 
9 

710 

1 

2 
3 
4 
5 
6 
7 
8 
9 

720 

1 

2 
3 

4 
5 
6 
7 
8 
9 

730 
1 
2 

3 
4 
5 
6 
7 
8 
9 

740 

1 
2 
3 

4 
5 
6 
7 
8 
9 

750 


84510  84516  84522  84528  84535  84541  84547  84553  84559  84560 

4572  4578  4584  4590  4597  4603  4609  4615  4621  4628 

4634  4640  4646  4652  4658  4665  4671  4677  4683  4689 

4696  4702  4708  4714  4720  4726  4733  4739  4745  4751 

4757  4763  4770  4776  4782  4788  4794  4800  4807  4813 

4819  4825  4831  4837  4844  4850  4856  4862  4868  4874 

4880  4887  4893  4899  4905  4911  4917  4924  4930  4936 

4942  4948  4954  4960  4967  4973  4979  4985  4991  4997 

5003  5009  5016  5022  5028  5034  5040  5046  5052  5058 

5065  5071  5077  5083  5089  5095  5101  5107  5114  5120 

85126  85132  85138  85144  85150  85156  85163  85169  85175  85181 

5187  5193  5199  5205  5211  5217  5224  5230  5236  5242 

5248  5254  5260  5266  5272  5278  . 5285  5291  5297  5303 

5309  5315  5321  5327  5333  5339  5345  5352  5358  5364 

5370  5376  5382  5388  5394  5400  5406  5412  5418  5425 

5431  5437  5443  5449  5455  5461  *467  5473  5479  5485 

5491  5497  5503  5509  5516  5522  5528  5534  5540  5546 

5552  5558  5564  5570  5576  55C2  5588  5594  5600  5606 

5612  5618  5625  5631  5637  5643  5649  5655  5661  5667 

5673  5679  6685  5691  5697  6703  5709  5715  5721  5727 

85733  85739  85745  85751  85757  85763  85769  85775  85781  85788 

5794  5800  5806  5812  5818  5824  5830  5836  5842  5848 

5854  5860  5866  5872  5878  5884  5890  5896  5902  5908 

5914  5920  5926  5932  5938  5944  5950  5956  5962  5968 

5974  5980  5986  5992  5998  6004  6010  6016  6022  6028 

6034  6040  6046  6052  6058  6064  6070  6076  6082  6088 

6094  6100  6106  6112  6118  6124  6130  6136  6141  6147 

6153  6159  6165  6171  6177  6183  6189  6195  6201  6207 

6213  6219  6225  6231  6237  6243  6249  6255  6261  6267 

6273  6279  6285  6291  6297  6303  6308  6314  6320  6326 

86332  86338  86344  86350  86356  86362  86368  86374  86380  86386 

6392  6398  6404  6410  6415  6421  6427  6433  6439  6445 

6451  6457  6463  6469  6475  6481  6487  6493  6499  6504 

6510  6516  6522  6528  6534  6540  6546  6552  6558  6564 

6570  6576  6581  6587  6593  6599  6605  6611  6617  6623 

6629  6635  6641  6646  6652  6658  6664  6670  6676  6682 

6688  6694  6700  6705  6711  6717  6723  6729  6735  6741 

6747  6753  6759  6764  6770  6776  6782  6788  6794  6800 

6806  6812  6817  6823  6829  6835  6841  6847  6853  6859 

6864  6870  6876  6882  6888  6894  6900  6906  6911  6917 

86923  86929  86935  86941  86947  86953  86958  86964  86970  86976 

6982  6988  6994  6999  7005  7011  7017  7023  7029  7035 

7040  7046  7052  7058  7064  7070  7075  7081  7087  7093 

7099  7105  7111  7116  7122  7128  7134  7140  7146  7151 

7157  7163  7169  7175  7181  7186  7192  7198  7204  7210 

7216  7221  7227  7233  7239  7245  7251  7256  7262  7268 

7274  7280  7286  7291  7297  7303  7309  7315  7320  7326 

7332  7338  7344  7349  7355  7361  7367  7373  7379  7384 

7390  7396  7402  7408  7413  7419  7425  7431  7437  7442 

7448  7454  7460  7466  7471  7477  7483  7489  7493  7500 

87506  87512  87518  87523  87529  87535  87541  87547  87552  87558 


TABLE  7.— LOGARITHMS  OF  NUMBERS.  171 


N0123456789 


750 
1 

2 
3 

4 
5 
6 
7 
8 
9 

760 
1 

2 
3 

4 
5 
6 

7 
8 
9 


3 
4 
5 
6 
7 
8 
9 

780 
1 
2 
3 
4 
5 
6 
7 
8 
9 

790 

1 
2 
3 
4 
5 
6 
7 
8 
9 

800 


87506  87512  87518  87523  87529  87535  87541  87547  87552  87558 

7564  7570  7576  7581  7587  7593  7599  7604  7610  7616 

7622  7628  7633  7639  7645  7651  7656  7662  7668  7674 

7679  7685  7691  7697  7703  .7708  7714  7720  7726  7731 

7737  7743  7749  7754  7760  7766  7772  7777  7783  7789 

7795  7800  7806  7812  7818  7823  7829  7835  7841  7846 

7852  7858  7864  7869  7875  7881  7887  7892  7898  7904 

7910  7915  7921  7927  7933  7938  7944  7950  7955  7961 

7967  7973  7978  7984  7990  7996  8001  8007  8013  8018 

8024  8030  8036  8041  8047.  8053  8058  8064  8070  8076 

88081  88087  88093  88098  88104  88110  88116  88121  88127  88133 

8138  8144  8150  8156  8161  8167  8173  8178  8184  8190 

8195  8201  8207  8213  8218  8224  8230  8235  8241  8247 

8252  8258  8264  8270  8275  8281  8287  8292  8298  8304 

8309  8315  8321  8326  8332  8338  8343  8349  8355  8360 

8366  8372  8377  8383  8389  8395  8400  8406  8412  8417 

8423  8429  8434  8440  8446  8451  8457  8463  8468  8474 

8480  8485  8491  8497  8502  8508  8513  8519  8525  8530 

8536  8542  8547  8553  8559  8564  8570  8576  8581  8587 

8593  8598  8604  8610  8615  8621  8627  8632  8638  8643 

88649  88655  88660  88666  88672  88677  88683  88689  88694  88700 

8705  8711  8717  8722  8728  8734  8739  8745  8750  8756 

8762  8767  8773  8779  8784  8790  8795  8801  8807  8812 

8818  8824  8829  8835  8840  8846  8852  8857  8863  8868 

8874  8880  8885  8891  8897  8902  8908  8913  8919  8925 

8930  8936  8941  8947  8953  S958  8964  8969  8975  8981 

8986  8992  8997  9003  9009  9014  9020  9025  9031  9037 

9042  9048  9053  9059  9064  9070  9076  9081  9087  9092 

9098  9104  9109  9115  9120  9126  9131  9137  9143  9148 

9154  9159  9165  9170  9176  9182  9187  9193  9198  9204 

89209  89215  89221  89226  89232  89237  89243  89248  89254  89260 

9265  9271  9276  9282  9287  9293  9298  9304  9310  9315 

9321  9326  9332  9337  9343  9348  9354  9360  9365  9371 

9376  9382  9387  9393  9398  1)404  9409  9415  9421  9426 

9432  9437  9443  9448  9454  9459  9465  9470  9476  9481 

9487  9492  9498  9504  9509  9515  9520  9526  9531  9537 

9542  9548  9553  9559  9564  9570  9575  9581  9586  9592 

9597  9603  9509  9614  9620  9625  9631  9636  9642  9647 

9853  9653  9664  9669  9675  9680  9686  9691  9697  9702 

9708  9713  9719  9724  9730  9735  9741  9746  9752  9757 

89763  89768  89774  89779  89785  89790  89796  89801  89807  89812 

9818  9323  9829  9834  9840  9845  9851  9856  9862  9867 

9873  9878  9883  9889  9894  9900  9905  9911  9916  9922 

9927  9933  9938  9944  9949  9955  9960  9966  9971  9977 

9982  9988  9993  9998900049000990015900209002690031 

90037  90042  90048  90053  0059  0064  0069  0073  0080  0086 

0091  0097  0102  0108  0113  0119  0124  0129  013$  0140 

0146  0151  0157  0162  0168  0173  0179  0184  0189  0195 

0200  0206  0211  0217  02'22  0227  0233  0238  0244  0249 

0255  0260  0266  0271  0276  0282  0287  0293  0298  0304 

90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 


172  TABLE  7.— LOGARITHMS  OF  NUMBERS. 


N0123456789 


800 

1 


4 
5 
6 
7 
8 
9 

810 

1 
2 
3 

4 
5 
6 
7 
8 
9 

820 

1 
2 
3 

4 
5 
6 
7 
8 
9 

830 

1 
2 
3 

4 
5 
6 
7 
8 
9 

840 
1 

2 
3 

4 
5 
6 

7 


850 


90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 

0363  0369  0374  0380  0385  0390  0396  0401  0407  0412 

0417  0423  0428  0434  0439  0445  0450  0455  0461  0466 

0472  0477  0482  0488  0493  0499  0504  0509  0515  0520 

0526  0531  0536  0542  0547  0553  0558  0563  0569  0574 

0580  0585  0590  0596  0601  0607  0612  0617  0623  0628 

0634  0639  0644  0650  0655  0660  0666  0671  0677  0682 

0687  0693  0698  0703  0709  0714  0720  0725  0730  0736 

0741  0747  0752  0757  0763  0768  0773  0779  0784  0789 

0795  0800  0806  0811  0816  0822  0827  0832  0838  0843 

90849  90854  90859  90865  90870  90875  90881  90886  90891  90897 

0902  0907  0913  0918  0924  0929  0934  0940  0945  0950 

0956  0961  0966  0972  0977  0982  0988  0993  0998  1004 

1009  1014  1020  1025  1030  1036  1041  1046  1052  1057 

1062  1068  1073  1078  1084  1089  1094  1100  1105  1110 

1116  1121  1126  1132  1137  1142  1148  1153  1158  1164 

1169  1174  1180  1185  1190  1196  1201  1206  1212  1217 

1222  1228  1233  1238  1243  1249  1254  1259  1265  1270 

1275  1281  1286  1291  1297  1302  1307  1312  1318  1323 

1328  1334  1339  1344  1350  1355  1360  1365  1371  1376 

91381  91387  91392  91397  91403  91408  91413  91418  91424  91429 

1434  1440  1445  1450  1455  1461  1466  1471  1477  1482 

1487  1492  1498  1503  1508  1514  1519  1524  1529  1535 

1540  1545  1551  1556  1561  1566  1572  1577  1582  1587 

1593  1598  1603  1609  1614  1619  1624  1630  1635  1640 

1645  1651  1656  1661  1666  1672  1677  1682  1687  1693 

1698  1703  1709  1714  1719  1724  1730  1735  1740  1745 

1751  1756  1761  1766  1772  1777  1782  1787  1793  1798 

1803  1808  1814  1819  1824  1829  1834  1840  1845  1850 

1855  1861  1866  1871  1876  1882  1887  1892  1897  1903 

91908  91913  91918  91924  91929  91934  91939  91944  91950  91955 

1960  1965  1971  1976  1981  1986  1991  1997  2002  2007 

2012  2018  2023  2028  2033  2038  2044  2049  2054  2059 

2065  2070  2075  2080  2085  2091  2096  2101  2106  2111 

2117  2122  2127  2132  2137  2143  2148  2153  2158  2163 

2169  2174  2179  2184  2189  2195  2200  2205  2210  2215 

2221  2226  2231  2236  2241  2247  2252  2257  2262  2267 

2273  2278  2283  2288  2293  2298  2304  2309  2314  2319 

2324  2330  2335  2340  2345  2350  2355  2361  2366  2371 

2376  2381  2387  2392  2397  2402  2407  2412  2418  2423 

92428  92433  92438  92443  92449  92454  92459  92464  92469  92474 

2480  2485  2490  2495  2500  2505  2511  2516  2521  2526 

2531  2536  2542  2547  2552  2557  2562  2567  2572  2578 

2583  2588  2593  2598  2603  2609  2614  2619  2624  2629 

2634  2639  2645  2650  2655  2660  2665  2670  2675  2681 

2686  2691  2696  2701  2706  2711  2716  2722  2727  2732 

2737  2742  2747  2752  2758  2763  2768  2773  2778  2783 

2788  2793  2799  2804  2809  2814  2819  2824  2829  2834 

2840  2845  2850  2855  2860  2865  2870  2875  2881  2886 

2891  2896  2901  2906  2911  2916  2921  2927  2932  2937 

92942  92947  92952  92957  92962  92967  92973  92978  92983  92988 


TABLE  7.— LOGARITHMS  OF  NUMBERS.  173 


NO123456789 


850 

1 
2 
3 

4 
5 
6 
7 
8 
9 

860 

1 
2 
3 
4 
5 
6 
7 
8 
9 

870 

1 

2 
3 
4 
5 
6 
7 
8 
9 

880 
1 
2 
3 
4 
5 
6 
7 
8 
9 

890 

1 
2 
3 
4 
5 
6 
7 
8 
9 

900 


92942  92947  92952  92957  92962  92967  92973  92978  92983  92988 

2993  2998  3003  3008  3013  3018  3024  3029  3034  3039 

3044  3049  3054  3059  3064  3069  3075  3080  3085  3090 

3095  3100  3105  3110  3115  3120  3125  3131  3136  3141 

3146  3151  3156  3161  3166  3171  3176  3181  3186  3192 

3197  3202  3207  3212  3217  3222  3227  3232  3237  3242 

3247  3252  3258  3263  3268  3273  3278  3283  3288  3293 

3298  3303  3308  3313  3318  3323  3328  3334  3339  3344 

3349  3354  3359  3364  3369  3374  3379  3384  3389  3394 

3399  3404  3409  3414  3420  3425  3430  3435  3440  3445 

93450  93455  93460  93465  93470*93475  93480  93485  93490  93495 

3500  3505  3510  3515  3520  3526  3531  3536  3541  3546 

3551  3556  3561  3566  3571  3576  3581  3586  3591  3596 

3601  3606  3611  3616  3621  3626  3631  3636  3641  3646 

3651  3656  3661  3666  3671  3676  3682  3687  3692  3697 

3702  3707  3712  3717  3722  3727  3732  3737  3742  3747 

3752  3757  3762  3767  3772  3777  3782  3787  3792  3797 

3802  3807  3812  3817  3822  3827  3832  3837  3842  3847 

3852  3857  3862  3867  3872  3877  3882  3887  3892  3897 

3902  3907  3912  3917  3922  3927  3932  3937  3942  3947 

93952  93957  93962  93967  93972  93977  93982  93987  93992  93997 

4002  4007  4012  4017  4022  4027  4032  4037  4042  4047 

4052  4057  4062  4067  4072  4077  4082  4086  4091  4096 

4101  4106  4111  4116  4121  4126  4131  4136  4141  4146 

4151  4156  4161  4166  4171  4176  4181  4186  4191  4196 

4201  4206  4211  4216  4221  4226  4231  4236  4240  4245 

4250  4255  4260  4265  4270  4275  4280  4285  4290  4295 

4300  4305  4310  4315  4320  4325  4330  4335  4340  434£ 

4349  4354  4359  4364  4369  4374  4379  4384  4389  4394 

4399  4404  4409  4414  4419  4424  4429  4433  4438  4443 

94448  94453  94458  94463  94468  94473  94478  94483  94488  94493 

4498  4503  4507  4512  4517  4522  4527  4532  4537  4542 

4547  4552  4557  4562  4567  4571  4576  4581  4586  4591 

4596  4601  4606  4611  4616  4621  4626  4630  4635  4640 

4645  4650  4655  4660  4665  4670  4675  4680  4685  4689 

4694  4699  4704  4709  4714  4719  4724  4729  4734  4738 

4743  4748  4753  4758  4763  4768  4773  4778  4783  4787 

4792  4797  4802  4807  4812  4817  4822  4827  4832  4836 

4841  4846  4851  4856  4861  4866  4871  4876  4880  4885 

4890  4895  4900  4905  4910  4915  4919  4924  4929  4934 

94939  94944  94949  94954  94959  94963  94968  94973  94978  94983 

4988  4993  4998  5002  5007  5012  5017  5022  5027  5032 

5036  5041  5046  5051  5056  5061  5066  5071  5075  5080 

5085  5090  5095  5100  5105  5109  5114  5119  5124  5129 

5134  5139  5143  5148  5153  5158  5163  5168  5173  5177 

5182  5187  5192  5197  5202  5207  5211  5216  5221  5226 

5231  5236  5240  5245  5250  5255  5260  5265  5270  5274 

5279  5284  5289  5294  5299  5303  5308  5313  5318  5323 

5328  5332  5337  5342  5347  5352  5357  5361  5366  5371 

5376  5381  5386  5390  5395  5400  5405  5410  5415  5419 

95424  95429  95434  95439  95444  95448  95453  95458  95463  95468 


174  TABLE  7.— LOGARITHMS  OF  NUMBERS. 


0123456789 


I 


900 

1 
2 
3 
4 
5 
6 
7 
8 
9 

910 

1 
2 
3 
4 
5 
6 
7 
8 
9 

920 
1 
2 
3 

4 
5 
6 

7 
8 
9 

930 

1 

2 
3 

4 
5 
6 
7 
8 
9 

940 

1 


4 
5 
6 
7 
8 
9 

950 


95424  95429  95434  95439  95444  95448  95453  95458  95463  95468 

5472  5477  6482  5487  5492  5497  5501  5506  5511  5516 

5521  5525  5530  5535  5540  5545  5550  5554  5559  5564 

5569  5574  5578  5583  5588  '  5593  5598  5602  5607  5612 

5617  5622  5626  5631  6636  5641  5646  5650  5655  5660 

5665  6670  5674  6679  6684  5689  5694  6698  5703  5708 

5713  5718  5722  5727  5732  5737  5742  5746  5751  5756 

5761  5766  5770  5775  5780  5785  5789  5794  5799  5804 

5809  5813  5818  5823  5828  5832  5837  5842  5847  6852 

5856  5861  6866  5871  5875  5880  5885  5890  5895  5899 

95904  95909  95914  95918  95923  95928  95933  95938  95942  95947 

5952  5957  5961  5966  5971  5976  5980  5985  5990  5995 

5999  6004  6009  6014  6019  6023  6028  6033  6038  6042 

6047  6052  6057  6061  6066  6071  6076  6080  6085  6090 

609^  6099  6104  6109  6114  6118  6123  6128  6133  6137 

6142  6147  6152  6156  6161  6166  6171  6175  6180  6185 

6190  6194  6199  6204  6209  6213  6218  6223  6227  6232 

6237  6242  6246  6251  6256  6261  6265  6270  6275  6280 

6284  6289  6294  6298  6303  6308  6313  6317  6322  6327 

6332  6336  6341  6346  6350  6355  6360  6365  6369  6374 

96379  96384  96388  96393  96398  96402  96407  96412  96417  96421 

6426  6431  6435  6440  6445  6450  6454  6459  6464  6468 

6473  6478  6483  6487  6492  6497  6501  6506  6511  6515 

6520  6525  6530  6534  6539  6544  6548  6553  6558  6562 

6567  6572  6577  6581  6586  6591  6595  6600  6605  6609 

6614  6619  6624  6628  6633  6638  6642  6647  6652  6656 

6661  6666  6670  6675  6680  6685  6689  6694  6699  6703 

6708  6713  6717  6722  6727  6731  6736  6741  6745  6750 

6755  6759  6764  6769  6774  6778  6783  6788  6792  6797 

6802  6806  6811  6816  6820  6825  6830  6834  6839  6844 

96848  96853  96858  96862  96867  96872  96876  96881  96886  96890 

6895  6900  6904  6909  6914  6918  6923  6928  6932  6937 

6942  6946  6951  6956  6960  6965  6970  6974  6979  6984 

6988  6993  6997  7002  7007  7011  7016  7021  7025  7030 

7035  7039  7044  7049  7053  7058  7063  7067  7072  7077 

7081  7086  7090  7095  7100  7104  7109  7114  7118  7123 

7128  7132  7137  7142  7146  7151  7155  7,160  7165  7169 

7174  7179  7183  7188  7192  7197  7202  7206  7211  7216 

7220  7225  7230  7234  7239  7243  7248  7253  7257  7262 

7267  7271  7276  7280  7285  7290  7294  7299  7304  7308 

97313  97317  97322  97327  97331  97336  97340  97345  97350  97354 

7359  7364  7368  7373  7377  7382  7387  7391  7396  7400 

7405  7410  7414  7419  7424  7428  7433  7437  7442  7447 

7451  7456  7460  7465  7470  7474  7479  7483  7488  7493 

7497  7502  7506  7511  7516  7520  7525  7529  7534  7539 

7543  7548  7552  7557  7562  7566  7571  7575  7580  7585 

7589  7594  7598  7603  7607  7612  7617  7621  7626  7630 

7635  7640  7644  7649  7653  7658  7663  7667  7672  7676 

7681  7685  -7690  7695  7699  7704  7708  7713  7717  7722 

7727  7731  7736  7740  7745  7749  7754  7759  7763  7768 

97772  97777  97782  97786  97791  97795  97800  97804  97809  97813 


TABLE  7.— LOGARITHMS   OF   NUMBERS.  175 


0123456789 


950  97772  97777  97782  97786  97791  97795  97800  97804  97809  97813 

1  7818  7823  7827  7832  7836  7841  7845  7850  7855  7859 

2  7864  7868  7873  7877  7882  7886  7891  7896  7900  7905 

3  7909  7914  7918  7923  7928  7932  7937  7941  7946  7950 

4  7955  7959  7964  7968  7973  7978  7982  7987  7991  7996 

5  8000  8005  8009  8014  8019  8023  8028  8032  8037  8041 

6  8046  8050  8055  8059  8064  8068  8073  8078  8082  8087 

7  8091  8096  8100  8105  8109  8114  8118  8123  8127  8132 

8  8137  8141  8146  8150  8155  8159  8164  8168  8173  8177 

9  8182  8186  8191  8195  8200.  8204  8209  8214  8218  8223 

960  98227  98232  98236  98241  98245  98250  98254  98259  98263  98268 

1  8272  8277  8281  8286  8290  8295  8299  8304  8308  8313 

2  8318  8322  8327  8331  8336  8340  8345  8349  8354  8358 

3  8363  8367  8372  8376  8381  8385  8390  8394  8399  8403 

4  8408  8412  8417  8421  8426  8430  8435  8439  8444  8448 

5  8453  8457  8462  8466  8471  8475  8480  8484  8489  8493 

6  8498  8502  8507  8511  8516  8520  8525  8529  8534  8538 

7  8543  8547  8552  8556  8561  8565  8570  8574  8579  8583 

8  8588  8592  8597  8601  8605  8610  8614  8619  8623  8628 

9  8632  8637  8641  8646  8650  8655  8659  8664  8668  8673 

970  98677  98682  98686  98691  98695  98700  98704  98709  98713  98717 

1  8722  8726  8731  8735  8740  8744  8749  8753  8758  8762 

2  8767  8771  8776  8780  8784  8789  8793  8798  8802  8807 

3  8811  8816  8820  8825  8829  8834  8838  8843  8847  8851 

4  8856  8860  8865  8869  8874  8878  8883  8887  8892  8896 

5  8900  8905  8909  8914  8918  8923  8927  8932  8936  8941 

6  8945  8949  8954  8958  8963  B967  8972  8976  8981  8985 

7  8989  8994  8998  9003  9007  9012  9016  9021  9025  9029 

8  9034  9038  9043  9047  9052  9056  9061  9065  9069  9074 

9  9078  9083  9087  9092  9096  9100  9105  9109  9114  9118 

980  99123  99127  99131  99136  99140  99145  99149  99154  99158  99162 

1  9167  9171  9176  9180  9185  9189  9193  9198  9202  9207 

2  9211  9216  9220  9224  9229  9233  9238  9242  9247  9251 

3  9255  9260  9264  9269  9273  9277  9282  9286  9291  9295 

4  9300  9304  9308  9313  9317  9322  9326  9330  9335  9339 

5  9344  9348  9352  9357  9361  9366  9370  9374  9379  9383 

6  9388  9392  9396  9401  9405  9410  9414  9419  9423  9427 

7  9432  9436  9441  9445  9449  9454  9458  9463  9467  9471 

8  9476  9480  9484  9489  9493  9498  9502  9506  9511  9516 

9  9520  9524  9528  9533  9537  9542  9546  9550  9555  9559 

990  99564  99568  99572  99577  99581  99585  99590  99594  99599  99603 

1  9607  9612  9616  9621  9625  9629  9634  9638  9642  9647 

2  9651  9656  9660  9664  9669  9673  9677  9682  9686  9691 

3  9695  9699  9704  9708  9712  9717  9721  9726  9730  9734 

4  9739  9743  9747  9752  9756'  9760  9765  9769  9774  9778 

5  9782  9787  9791  9795  9800  9804  9808  9813  9817  9822 
i  9826  9830  9835  9839  9843  9848  9852  9856  9861  9865 

7  9870  9874  9878  9883  9887  9891  9896  9900  9904  9909 

8  9913  9917  9922  9926  9930  9935  9939  9944  9948  9952 

9  9957  9961  9965  9970  9974  9978  9983  9987  9991  9996 

1000  00000  00004  00009  00013  00017  00022  00026  00030  00035  00039 


176     TABLE  8.— LOGARITHMIC   SINES  AND  COSINES. 


t 

0° 

1° 

2° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

—00 

10.00000 

8.24186 

9.99993 

8.54282 

9  99974 

60 

1 

6.46373 

00000 

24903 

99993 

54642 

99973 

59 

2 

76476 

00000 

25609 

'  99993 

54999 

99973 

58 

3 

94085 

00000 

26304 

99993 

55354 

99972 

57 

4 

7.0G579 

00000 

26988 

99992 

55705 

99972 

56 

5 

16270 

00000 

27661 

99992 

56054 

69971 

55 

6 

24188 

00000 

28324 

99992 

56400 

99971 

54 

7 

30882 

00000 

28977 

99992 

56743 

99970 

51 

8 

36682 

00000 

29621 

99992 

57084 

99970 

52 

9 

41797 

00000 

30255 

99991 

57421 

99969 

51 

10 

7.46373 

10.00000 

8.30879 

9.99991 

8.57757 

9.99969 

50 

11 

50512 

00000 

31495 

99991 

58089 

99968 

49 

12 

54291 

00000 

32103 

99990 

58419 

99968 

48 

13 

57767 

00000 

32702 

99990 

58747 

99967 

47 

14 

60985 

00000 

33292 

99990 

59072 

99967 

46 

15 

63982 

00000 

33875 

99990 

59395 

99967 

45 

16 

66784 

00000 

34450 

99989 

59715 

99960 

44 

17 

69417 

9.99999 

35018 

99989 

60033 

99966 

43 

18 

71900 

99999 

35578 

99989 

60349 

99965 

42 

19 

74248 

99999 

36131 

99989 

60662 

99964 

41 

20 

7.76475 

9.99999 

8.36678 

9.99988 

8.60973 

9.99964 

40 

21 

78594 

99999 

37217 

99988 

61282 

99963 

39 

22 

80615 

99999 

37750 

99988 

61589 

99963 

38 

23 

82545 

99999 

38276 

99987 

61894 

99962 

37 

24 

84393 

99999 

38796 

99987 

62196 

99962 

36 

25 

86166 

99999 

39310 

99987 

62497 

99961 

35 

26 

87870 

99999 

39818 

99986 

62795 

99961 

34 

27 

89509 

99999 

40320 

99986 

63091 

99960 

33 

28 

91088 

99999 

40816 

99986 

63385 

99960 

32 

29 

92612 

99998 

41307 

99985 

63678 

99959 

31 

30 

7.94084 

9.99998 

8.41792 

9.99985 

8.63968 

9.99959 

30 

31 

95508 

99998 

42272 

99982 

64256 

99958 

29 

32 

96887 

99998 

42746 

99984 

64543 

99958 

28 

33 

98223 

99998 

43216 

99984 

64827 

99957 

27 

34 

99520 

99998 

43680 

99984 

65110 

99956 

26 

35 

8.00779 

99998 

44139 

99983 

65391 

99956 

25 

36 

02002 

99998 

44594 

99983 

65670 

99955 

24 

37 

03192 

99997 

45044 

99983 

65947 

99955 

23 

38 

04350 

99997 

45489 

99982 

66223 

99954 

22 

39 

05478 

99997 

45930 

99982 

66497 

99954 

21 

40 

8.06578 

9.99997 

8.46366 

9.99982 

8.66769 

9.99953 

20 

41 

07650 

99997 

46799 

99981 

67039 

99952 

19 

42 

08696 

99997 

47226 

99981 

67308 

99952 

18 

43 

09718 

99997 

47650 

99981 

67575 

99951 

17 

44 

10717 

99996 

48069 

99980 

67841 

99951 

16 

45 

11693 

99996 

48485 

99980 

68104 

99950 

15 

46 

12647 

99996 

48896 

99979 

68367 

99949 

14 

47 

13581 

99996 

49304 

99979 

68627 

99949 

13 

48 

14495 

99996 

49708 

99979 

68886 

99948 

12 

49 

15391 

99996 

50108 

99978 

69144 

99948 

11 

50 

8.16268 

9.99995 

8.50504 

9.99978 

8.69400 

9.99947 

10 

51 

17128 

99995 

50897 

99977 

69654 

99946 

9 

52 

17971 

99995 

51287 

99977 

69907 

99946 

8 

53 

18798 

99995 

51673 

99977 

70159 

99945 

7 

54 

19610 

99995 

52055 

99976 

70409 

99944 

6 

55 

20407 

99994 

52434 

99976 

70658 

99944 

5 

56 

21189 

99994 

52810 

99975 

70905 

99943 

4 

57 

21958 

99994 

53183 

99975 

71151 

99942 

3 

58 

22713 

99994 

53552 

99974 

71395 

99942 

2 

59 

23456 

99994 

53919 

99974 

71638 

99941 

1 

60 

24186 

99993 

54282 

99974 

71880 

99940 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

89° 

88 

0 

87° 

TABLE  8.— LOGARITHMIC   SINES  AND  COSINES.    177 


/ 

3°             4°             6° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

8.71880 

9.99940 

8.84358 

9.99894 

8.94030 

9.99834 

60 

1 

72120 

99940 

84539 

99893 

94174 

99833 

59 

2 

72359 

99939 

84718 

99892 

94317 

99832 

58 

3 

72597 

99938 

84897 

99891 

944C1 

99831 

57 

4 

72834 

99938 

85075 

99891 

94603 

99830 

66 

5 

73069 

99937 

85252 

99890 

94746 

99829 

55 

6 

73303 

99936 

85429 

99889 

94887 

998*8 

54 

7 

73535 

99936 

85605 

99888 

95029 

99827 

53 

8 

73767 

99935 

85780 

99887 

95170 

99825 

52 

9 

73997 

99934 

85955 

99886 

95310 

99824 

51 

10 

8.74226 

9.99934 

8.86128 

9.99885 

8.95450 

9.99823 

50 

11 

74454 

99933 

86301 

99.884 

95589 

99822 

49 

12 

74680 

99932 

86474 

99*883 

95728 

99821 

48 

13 

74906 

99932 

86645 

99882 

95867 

99820 

47 

14 

75130 

99931 

86816 

99881 

96005 

99819 

46 

15 

75853 

99930 

86987 

99880 

96143 

99817 

45 

16 

75575 

99929 

87156 

99879 

96280 

99816 

44 

17 

75795 

99929 

87325 

99879 

96417 

99815 

43 

18 

76015 

99928 

87494 

99878 

96553 

99814 

42 

19 

76234 

99927 

87661 

99877 

96689 

99813 

41 

20 

8.76451 

9.99926 

8.87829 

9.99876 

8.96825 

9.99812 

40 

21 

76667 

99926 

87995 

99875 

96960 

99810 

39 

22 

76883 

99925 

88161 

99874 

97095 

99809 

38 

23 

77097 

99924 

88326 

99873 

97229 

99808 

37 

24 

77310 

99923 

88490 

99872 

97363 

99807 

36 

25 

77522 

99923 

88654 

99871 

97496 

99806 

35 

26 

77733 

99922 

88817 

99870 

97629 

99804 

34 

27  , 

77943 

99921 

88980 

99869 

97762 

99803 

33 

28 

78152 

99920 

89142 

99868 

97894 

99802 

33 

29 

78360 

99920 

89304 

99867 

98026 

99801 

31 

30 

8.78568 

9.99919 

8.89464 

9.99866 

8.98157 

9.99800 

30 

31 

78774 

99918 

89625 

99805 

98288 

99798 

29 

32 

78979 

99917 

89784 

99864 

98419 

99797 

28 

33 

79183 

99917 

89943 

99863 

98549 

99796 

27 

34 

79386 

99916 

P0102 

9986'* 

98679 

99795 

26 

35 

79588 

99915 

90260 

99861 

98808 

99793 

25 

36 

79789 

99914 

90417 

99860 

98937 

99792 

24 

37 

79990 

99913 

90574 

99859 

99066 

99791 

23 

38 

80189 

99913 

90730 

99858 

99194 

99790 

22 

39 

80388 

99912 

90885 

99857 

99322 

99788 

21 

40 

8.80585 

9.99911 

8.91040 

9.99856 

8.99450 

9.99787 

20 

41 

8078-2 

99910 

91195 

99855 

99577 

99786 

19 

42 

80978 

99909 

91349 

99854 

99704 

99785 

18 

43 

81173 

99909 

91502 

99853 

99830 

99783 

17 

44 

81367 

99908 

91655 

99852 

99956 

99782 

16 

45 

81560 

99907 

91807 

99851 

9.00082 

99781 

15 

46 

81752 

99906 

91959 

99850 

00207 

99780 

14 

47 

81944 

99905 

92110 

99848 

00332 

99778 

13 

48 

82134 

99904 

92261 

99847 

00456 

99777 

12 

49 

82324 

99904 

92411 

99846 

00581 

99776 

11 

50 

8.82513 

9.99903 

8.92561 

9.99845 

9.00704 

9.99775 

10 

51 

82701 

99902 

92710 

99844 

00828 

99773 

9 

52 

82888 

99901 

92859 

99843 

00951 

99772 

8 

53 

83075 

99900 

93007 

99842 

01074 

99771 

7 

54 

83261 

99899 

93154 

99841 

01196 

99769 

6 

55 

83446 

99898 

93301 

99840 

01318 

99768 

5 

56 

83630 

99898 

93448 

99839 

01440 

99767 

4 

57 

83813 

99897 

93594 

99838 

01561 

99765 

3 

58 

83996 

99896 

93740 

99837 

01682 

99764 

2 

59 

84177 

99895 

93885 

99836 

01803 

99763 

1 

60 

84358 

99894 

94030 

99834 

01923 

99761 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

i 

86° 

85° 

84° 

178     TABLE  8.— LOGARITHMIC  SINES  AND  COSINES. 


/ 

6° 

7° 

8» 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.01923 

9.99761 

9.08589 

9.99675 

9.14356 

9.99575 

60 

1 

02043 

99760 

08692 

99674 

14445 

99574 

59 

2 

02163 

99759 

08795 

99672 

14535 

99572  • 

58 

3 

02283 

99757 

08897' 

99670 

14624 

99570 

57 

4 

03402 

99756 

08999 

99669 

14714 

99568 

56 

5 

02520 

99755 

09101 

99667 

14803 

99566 

55 

6 

02639 

99753 

09202 

99666 

14891 

99565 

54 

7 

02757 

99752 

09304 

99664 

14980 

99563 

53 

8 

02874 

99751 

09405 

99663 

15069 

99561 

52 

9 

02992 

99749 

09506 

99661 

15157 

99559 

51 

10 

9.03109 

9.99748 

9.09606 

9.99659 

9.15245 

9.99557 

50 

11 

03226 

99747 

09707 

99658 

15333 

99556 

49 

12 

03342 

99745 

09807 

99656 

15421 

99554 

48 

13 

03458 

99744 

09907 

99655 

15508 

99552 

47 

14 

03574 

99742 

10006 

99653 

15596 

99550 

46 

15 

03690 

99741 

10106 

99651 

15683 

99548 

45 

16 

03805 

99740 

10205 

99650 

15770 

99546 

44 

17 

03920 

99738 

10304 

99648 

15857 

99545 

43 

18 

04034 

99737 

10402 

99647 

15944 

99543 

42 

19 

04149 

99736 

10501 

99645 

1C030 

99541 

41 

20 

9.04262 

9.99734 

9.10599 

9.99643 

9.16116 

9.99539 

40 

21 

04376 

99733 

10697 

99642 

16203 

99537 

39 

22 

04490 

99731 

10795 

99640 

16289 

99535 

38 

23 

04603 

99730 

10893 

99638 

16374 

89533 

37 

24 

04715 

99728 

10990 

99637 

16460 

99532 

86 

25 

04828 

99727 

11087 

99635 

16545 

99530 

35 

26 

04940 

99726 

11184 

99633 

16631 

99528 

34 

27 

05052 

99724 

11281 

99632 

16716 

99526 

33 

28 

05164 

99723 

11377 

99630 

16801 

99524 

32 

29 

05275 

99721 

11474 

99629 

16886 

99522 

31 

30 

9.05386 

9.99720 

9.11570 

9.99627 

9.16970 

9.99520 

30 

31 

05497 

99718 

11666 

99625 

17055 

99518 

29 

32 

05607 

99717 

11761 

99624 

17139 

99517 

28 

33 

05717 

99716 

11857 

99622 

17223 

99515 

27 

34 

05827 

99714 

11952 

99620 

J7807 

99513 

26 

35 

05937 

99713 

12047 

99618 

17391 

99511 

25 

36 

06046 

99711 

12142 

99617 

17474 

99509 

24 

37 

06155 

99710 

12236 

99615 

17558 

99507 

23 

38 

06264 

99708 

1233i 

99613 

17641 

99505 

22 

39 

06372 

99707 

12425 

99612 

17724 

99503 

21 

40 

9.06481 

9.99705 

9.12519 

&.  99610 

9.17807 

9.99501 

20 

41 

06589 

99704 

12612 

99608 

17890 

99499 

19 

42 

06696 

99702 

12706 

99607 

17973 

99497 

18 

43 

06804 

99701 

12799 

99605 

18055 

99495 

17 

44 

06911 

99699 

12892 

99603 

18137 

99494 

16 

45 

07018 

99698 

12985 

99601 

18220 

99492 

15 

46 

07124 

99696 

13078 

99600 

18302 

99490 

14 

47 

07231 

99695 

13171 

99598 

18383 

99488 

13 

48 

07337 

99G93 

13263 

99596 

18465 

99486 

19 

49 

07442 

99692 

13355 

99595 

18547 

99484 

11 

50 

9.07548 

9.99690 

9.13447 

9.99593 

9.18628 

9.99482 

30 

51 

07653 

99689 

13539 

99591 

18709 

99480 

9 

52 

07758 

99687 

13630 

99589 

18790 

99478 

8 

53 

07863 

99686 

13722 

99588 

18871 

99476 

7 

54 

07968 

99684 

13813 

99586 

18952 

99474 

6 

55 

08072 

99683 

18904 

99584 

19033 

99472 

5 

56 

08176 

99681 

13994 

99582 

19113 

99470 

4 

57 

08280 

99680 

14085 

99581 

19193 

99468 

3 

58 

08383 

99678 

14175 

99579 

19273 

99466 

2 

59 

08486 

99677 

14266 

99577 

19353 

99464 

1 

60 

08589 

99675 

14356 

99575 

19433 

99462 

0 

I  » 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

I 

88° 

82° 

81* 

TABLE  8. —LOGARITHMIC   SINES  AND  COSINES,  179 


/ 

9 

10° 

11 

» 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.19433 

S.  99482 

9.23967 

9.99335 

9.28060 

9.99195 

60 

1 

19513 

99460 

24039 

99333 

28125 

99192 

59 

2 

19592 

99458 

24110 

99331 

28190 

99190 

58 

3 

19672 

99456 

24181 

99328 

28254 

99187 

57 

4 

19751 

99454 

24253 

99326 

28319 

99185 

56 

5 

19830 

9945* 

24324 

99324 

28384 

99182 

55 

6 

19909 

99450 

24395 

99322 

28448 

99180 

54 

7 

19988 

99448 

24466 

99319 

28512 

99177 

53 

8 

20067 

99446 

24536 

99317 

28577 

99175 

52 

9 

20145 

99444 

24607 

99315 

28641 

99172 

61 

10 

9.20223 

9.99442 

9.24677 

9.99313 

9.28705 

9.99170 

50 

11 

20302 

99440 

24748 

99310 

2876» 

99167 

49 

12 

20380 

99438 

24818 

99308 

28833 

99165 

48 

13 

20458 

99436 

24888 

99306 

28896 

99162 

47 

14 

20535 

99434 

24958 

99304 

28960 

99160 

46 

15 

20613 

99432 

25028 

99301 

29024 

99157 

45 

16 

20691 

99429 

25098 

99299 

29087 

99155 

44 

17 

207G8 

99427 

25168 

99297 

29150 

99152 

43 

18 

20845 

99425 

25237 

99294 

29214 

99150 

42 

19 

20922 

99423 

25307 

99292 

29277 

99147 

41 

20 

9.20999 

9.99421 

9.25376 

9.99290 

9.29340 

9.99145 

40 

21 

21076 

99419 

25445 

99288 

29403 

99142 

39 

22 

21153 

99417 

-25514 

99285 

29466 

99140 

38 

23 

21229 

99415 

25583 

99283 

29529 

99137 

37 

24     21306 

99413 

25652 

99281 

29591 

99135 

36 

25 

21382 

99411 

25721 

99278 

29654 

99132 

35 

2o 

21458 

99409 

25790 

99276 

29716 

99130 

34 

27 

21534 

99407 

25858 

99274 

29779 

99127 

33 

28 

21610 

99404 

25927 

99271 

29841 

99124 

32 

29 

21685 

99402 

25995 

99269 

29903 

99122 

31 

30 

9.21761 

9.99400 

9.26063 

9.99267 

9.29966 

9.99119 

30 

31 

21836 

99398 

26131 

99264 

30028 

99117 

29 

32 

21912 

99396 

26199 

99262 

30090 

99114 

28 

33 

21987 

99394 

26267 

99260 

30151 

99112 

27 

34 

22062 

99392 

26335 

99257 

30213 

99109 

26 

35 

22137 

99390 

26403 

99255 

30275 

99106 

25 

36 

22211 

99388 

26470 

99252 

30336 

99104 

24 

37 

22286 

99385 

26538 

99250 

30398 

99001 

23 

38 

22361 

99383 

26605 

99248 

30459 

99099 

22 

39 

22435 

99381 

26672 

99245 

30521 

99096 

21 

40 

9.22509 

9.99379 

9.26739 

9.99243 

9.30582 

9.99093 

20 

41 

22583 

99377 

26806 

99241 

30643 

99091 

19 

42 

22657 

99375 

26873 

99238 

30704 

99088 

18 

43 

22731 

99372 

26940 

99236 

30765 

99086 

17 

44 

22805 

99370 

27007 

99233 

30826 

99083 

16 

45 

22878 

99368 

27073 

99231 

30887 

99080 

15 

46 

22952 

99366 

27140 

99229 

30947 

99078 

14 

47 

23025 

99364 

27206 

99226 

31008 

99075 

13 

48 

23098 

99362 

27273 

99224 

31068 

99072 

12 

49 

23171 

99359 

27339 

99221 

81129 

99070 

11 

50 

9.23244 

9.99357 

9.27405 

9.99219 

9.31189 

9.99067 

10 

51 

23317 

99355 

27471 

99217 

31250 

99064 

9 

52 

23390 

99353 

27537 

99214 

31310 

99062 

8 

53 

23462 

99351 

27602 

99212 

31370 

99059 

7 

54 

23535 

99348 

27668 

99209 

31430 

99056 

6 

55 

23607 

99346 

27734 

99207 

31490 

99054 

5 

56 

23679 

99344 

27799 

99204 

.  31549 

99051 

4 

57 

23752 

9934? 

27864 

99202 

31609 

99048 

3 

58 

23823 

99340 

27930 

99200 

31669 

99046 

2 

59 

23895 

99337 

27995 

99197 

31728 

99043 

1 

60 

23967 

99335 

28060 

99195 

31788 

99040 

0 

t 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

80° 

79° 

78° 

180     TABLE  8. —LOGARITHMIC   SINES  AND  COSINES. 


I 

2° 

13° 

14° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.31788 

9.99040 

9.35209 

9.98872 

9.38368 

9.98690 

60 

1 

31847 

99038 

35263 

98869 

38418 

98687 

59 

2 

31907 

99035 

35318 

98867 

38469 

98684 

58 

3 

31966 

99032 

35373 

98864 

38519 

98681 

57 

4 

32025 

99030 

35427 

98861 

38570 

98678 

56 

5 

32084 

99027 

35481 

98858 

38620 

98675 

55 

6 

32143 

99024 

35536 

98855 

38670 

98671 

54 

7 

32202 

99022 

35590 

98852 

38721 

98668 

53 

8 

32261 

99019 

35644 

98849 

38771 

98665 

52 

9 

32319 

99016 

35698 

98846 

38821 

98662 

51 

10 

9.32378 

9.99013 

9.35752 

9.98843 

9.38871 

9.98659 

50 

11 

32437 

99011 

35806 

98840 

38921 

98656 

49 

12 

32495 

99008 

35860 

98837 

38971 

98652 

48 

13 

32553 

99005 

35914 

98834 

39021 

98649 

47 

14 

32612 

99002 

35968 

98831 

39071 

98646 

46 

15 

32670 

99000 

36022 

98828 

39121 

98643 

45 

16 

32728 

98997 

36075 

98825 

39170 

98640 

44 

17 

32786 

98994 

'  36129 

98822 

39220 

98636 

43 

18 

32844 

98991 

36182 

98819 

39270 

98633 

42 

19 

32902 

98989 

36236 

98816 

39319 

98030 

41 

20 

9.32960 

9.98986 

9.36289 

9.98813 

9.39369 

9.98627 

40 

21 

33018 

98983 

36342 

98810 

39418 

98623 

39 

22 

33075 

98980 

36395 

98807 

39467 

98620 

38 

23 

33133 

98978 

36449 

98804 

39517 

98617 

37 

24 

33190 

98975 

36502 

98801 

39566 

98614 

36 

25 

33248 

98972 

36555 

98798 

39615 

98610 

35 

26 

33305 

98969 

36608 

98795 

39664 

98607 

34 

27 

33362 

98967 

36660 

98792 

39713 

98604 

33 

28 

33420 

98964 

36713 

98789 

39762 

98601 

32 

29 

33477 

98961 

36766 

98786 

39811 

98597 

31 

30 

9.33534 

9.98958 

9.36819 

9.98783 

9.39860 

9.98594 

30 

31 

33591 

98955 

36871 

98780 

39909 

98591 

29 

32 

33647 

98953 

36924 

98777 

39958 

98588 

28 

33 

33704 

98950 

36976 

98774 

40006 

98584 

27 

34 

33761 

98947 

37028 

98771 

40055 

98581 

26 

35 

33818 

98944 

37081 

98768 

40103 

98578 

25 

36 

33874 

98941 

37133 

98765 

40152 

98574 

24 

37 

83931 

98938 

37185 

98762 

40200 

98571 

23 

38 

33987 

98936 

37237 

98759 

40249 

98568 

22 

39 

34043 

98933 

37289 

98756 

40297 

98565 

21 

40 

9.34100 

9.98930 

9.37341 

9.98753 

9.40346 

9.98561 

20 

41 

34156 

98927 

37393 

98750 

40394 

98558 

19 

42 

34212 

98924 

37445 

98746 

40442 

98555 

18 

43 

34268 

98921 

37497 

98743 

40490 

98551 

17 

44 

34324 

98919 

37549 

98740 

40538 

98548 

16 

45 

34380 

98916 

37600 

98737 

4058G 

98545 

15 

46 

34436 

98913 

37652 

98734 

40634 

98541 

14 

47 

34491 

98910 

37703 

98731 

40682 

98538 

13 

48 

34547 

98907 

37755 

98728 

40730 

98535 

12 

49 

34602 

98904 

37806 

98725 

40778 

98531 

11 

50 

9.34658 

9.98901 

9.37858 

9.98722 

9.40825 

9.98528 

10 

51 

34713 

98898 

37909 

98719 

40873 

98525 

9 

52 

34769 

98896 

37960 

'  98715 

40921 

98521 

8 

53 

34824 

98893 

38011 

98712 

40968 

98518 

7 

54 

34879 

98890 

38062 

98709 

41016 

98515 

6 

55 

34934 

98887 

38113 

98706 

41063 

98511 

5 

56 

34989 

98884 

38164 

98703 

41111 

98508 

4 

57 

35044 

98881 

38215 

98700 

41158 

98505 

3 

58 

35099 

98878 

38266 

98697 

41205 

98501 

2 

59 

35154 

98875 

38317 

98694 

41252 

98498 

1 

00 

35209 

98872 

38368 

98690 

41300 

98494 

0 

t 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

t 

77o 

76° 

75° 

TABLE  8.— LOGARITHMIC   SINES  AND  COSINES.    181 


t 

15° 

16° 

17° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.41300 

9.98494 

9.44034 

9.98284 

9.46594 

9.98060 

60 

I 

41347 

98491 

44078 

98281 

46635 

98056 

59 

2 

41394 

98488 

44122 

98277 

46676 

98052 

58 

3 

41441 

98484 

44166 

98273 

46717 

98048 

57 

4 

41488 

98481 

44210 

98270 

46758 

98044 

56 

5 

41535 

98477 

44253 

98266 

46800 

98040 

55 

6 

41582 

98474 

44297 

98262 

46841 

98036 

54 

7 

416-,>8 

98471 

44341 

98259 

46882 

98032 

53 

8 

41675 

98467 

44385 

98255 

46923 

98029 

52 

9 

41722 

98464 

44428 

98251 

46964 

98025 

51 

10 

9.41768 

9.98460 

9.44472 

9.98248 

9.47005 

9.98021 

50 

11 

41815 

98457 

44516 

98244 

47045 

98017 

49 

12 

41861 

98453 

44559 

96240 

47086 

98013 

48 

18 

41908 

98450 

44602 

98237 

47127 

98009 

47 

14 

41954 

98447 

44646 

98233 

47168 

98005 

46 

15 

42001 

98443 

44689 

98229 

47209 

98001 

45 

16 

42047 

98440 

44733 

98226 

47249 

97997 

44 

17 

42093 

98436 

44776 

98222 

47290 

97993 

43 

18 

42140 

98433 

44819 

98218 

47330 

97989 

42 

19 

42186 

98429 

44862 

98215 

47371 

97986 

41 

20 

9.42232 

9.98426 

9.44905 

9.98211 

9.47411 

9.97982 

40 

21 

42278 

98422 

44948 

98207 

47452 

97978 

39 

22 

42324 

98419 

44992 

98204 

47492 

97974 

38 

23 

42370 

98415 

45035 

98200 

47533 

97970 

37 

24 

42416 

98412 

45077 

98196 

47573 

97966 

36 

25 

42461 

98409 

45120 

98192 

47613 

97962 

35 

26 

42507 

98405 

45163 

98189 

41654 

97958 

34 

27 

42553 

98402 

45206 

98185 

47694 

97954 

33 

28 

42599 

98398 

45249 

98181 

47734 

97950 

32 

29 

42(344 

98395 

45292 

98177 

47774 

97946 

31 

30 

9.42690 

9.98391 

9.45334 

9.98174 

9.47814 

9.97942 

30 

31 

42735 

98388 

45377 

98170 

47854 

97938 

29 

32 

42781 

98384 

45419 

98166 

47894 

97934 

28 

33 

42826 

98381 

45462 

98162 

47934 

97930 

27 

34 

42872 

98377 

45504 

98159 

47974 

97926 

26 

35 

42917 

98373 

45547 

98155 

4h014 

97922 

25 

36 

42962 

98370 

45589 

98151 

48054 

97918 

24 

37 

4:3008 

98366 

45632 

98147 

48094 

97914 

23 

38 

43053 

98363 

45674 

98144 

48133 

97910 

22 

39 

43098 

98359 

45716 

98140 

48173 

97906 

21 

40 

9.43143 

9.98356 

9.45758 

9.98136 

9.48213 

9.97902 

20 

41 

43188 

98352 

45801 

98132 

48252 

97898 

19 

42 

43233 

98349 

45843 

98129 

48292 

97894 

18 

43 

43278 

98345 

4.1885 

98125 

48332 

97890 

17 

44 

43323 

98342 

45927 

98121 

48371 

97886 

16 

45 

43367 

98338 

45969 

98117 

48411 

97882 

15 

46 

43412 

98334 

46011 

98113 

48450 

97878 

14 

47 

43457 

98331 

46053 

98110 

48490 

97874 

18 

48 

43502 

98327 

46095 

98106 

48529 

97870 

12 

49 

43546 

98324 

46136 

98102 

48568 

97866 

|1 

50 

9.43591 

9.98320 

9.46178 

9.98098 

9.48607 

9.97861 

10 

51 

43635 

98317 

46220 

98094 

48647 

97857 

9 

52 

43680 

98313 

46-^62 

98090 

48686 

97853 

8 

53 

43724 

98309 

46303 

9H087 

48725 

97849 

7 

54 

43769 

98306 

46345 

98083 

48764 

97845 

6 

55 

43813 

98302 

46386 

98079 

48803 

97841 

5 

56 

43857 

98299 

46428 

98075 

48842 

97837 

4 

57 

43901 

98295 

46469 

98071 

48881 

97833 

3 

58 

43946 

98291 

46511 

98067 

4S920 

97829 

2 

59 

43990 

98288 

46552 

98063 

48959 

97825 

1 

60 

44034 

98284 

36594 

98060 

48998 

97821 

0 

f 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

74° 

73° 

72° 

182     TABLE  8.— LOGARITHMIC   SINES  AND  COSINES. 


18° 

19° 

20° 

Sine 

Cosine 

Sine 

Cosine 

Sine    Cosine 

0 

9.48998 

9.97821 

9.51264 

9.97567 

9.53405   9 

97299 

60  1 

1 

49037 

97817 

51301 

97563 

53440 

97294 

59  ) 

2 

49076 

97812 

51338 

97558 

53475 

97289 

58 

3 

49115 

97808 

51374 

97554 

53509 

97285 

57 

4 

49153 

97804 

51411 

97550 

53544 

97280 

50 

5 

49192 

97800 

51447 

97545 

53578 

97276 

55 

6 

49231 

97796 

51484 

97541 

53613 

97271 

54 

7 

49269 

97792 

51520 

97536 

53647 

97266 

53 

8 

49308 

97788 

51557 

97532 

53682 

97262 

52 

9 

49347 

97784 

51593 

97528 

53716 

97257 

51 

10 

9.49385 

9.97779 

9.51629 

9.97523 

9.53751   9 

97252 

50 

11 

49424 

97775 

51666 

97519 

53785 

97248 

49 

12 

49462 

97771 

51702 

97515 

53819 

97243 

48 

18 

49500 

97767 

51738 

97510 

53854 

97238 

47 

14 

49539 

97763 

51774 

97506 

53888 

97234 

46 

15 

49577 

97759 

51811 

97501 

53922 

97229  ; 

45 

16 

49615 

97754 

51847 

97497 

53957 

97224 

44 

17 

49654 

97750 

51883 

97492 

53991 

97220 

43 

18 

49692 

97746 

51919 

97488 

54025 

97215 

42 

19 

49730 

97742 

51955 

97484 

54059 

97210 

41 

20 

9.49768 

9.97738 

9.51991 

9.97479 

9.54093   9 

97206 

40 

21 

49806 

97734 

52027 

97475 

54127 

97201 

39 

22 

49844 

97729 

52063 

97470 

54161 

97196 

38 

23 

49882 

97725 

52099 

97466 

54195 

97192 

37 

24 

49920 

97721 

52135 

97461 

54229 

97187 

36 

25 

49958 

97717 

52171 

97457 

54263 

97182 

35 

26 

49996 

97713 

52207 

97453 

54297 

97178 

34 

27 

50034 

97708 

52242 

97448 

54331 

97173 

33 

28 

50072 

97704 

52278 

97444 

54365 

97168 

32 

29 

50110 

97700 

52314 

97439 

54399 

97163 

31 

30 

9.50148 

9.97696 

9.52350 

9.97435 

9.54433   9 

97159 

30 

31 

50185 

97691 

52385 

97430 

54466 

97154 

29 

32 

50223 

97687 

52421 

97426 

54500 

97149 

28 

33 

50261 

97683 

52456 

97421 

54534 

97145 

27 

34 

50298 

97679 

52492 

97417 

54567 

97140 

26 

35 

50336 

97674 

52527 

97412 

54601 

97135 

25 

36 

50374 

97670 

52563 

97408 

54635 

97130 

24 

37 

50411 

97666 

52598 

97403 

54668 

97126 

23 

38 

50449 

97662 

52634 

97399 

54702 

97121 

22 

39 

50486 

97657 

52669 

97394 

54735 

97116 

21 

40 

9.50523 

9.97653 

9.52705 

9.97390 

9.54769   9 

97111 

20 

41 

50561 

97649 

52740 

97385 

54802  . 

97107 

19 

42 

50593 

97645 

52775 

97381 

54836 

97102 

18 

43 

50635 

97640 

52811 

97376 

54869 

97097 

17 

44 

50673 

97636 

52846 

97372 

54903 

97092 

16 

45 

50710 

97632 

52881 

97367 

54936 

97087 

15 

46 

50747 

97628 

52916 

97363 

54969 

97083 

14 

47 

50784 

97623 

52951 

97358 

55003 

97078 

13 

48 

50821 

97619 

52986 

97353 

55036 

97073 

12 

49 

50858 

97615 

53021 

97349 

55069 

97068 

11 

50 

9.50896 

9.97610 

9.53056 

9.97344 

9.55102   9 

.97063 

10 

51 

50933 

97606 

53092 

97340 

55136 

97059 

9 

52 

50970 

97602 

53126 

97335 

55169 

97054 

8 

53 

51007 

97597 

53161 

97331 

55202 

97049 

7 

54 

51043 

97593 

53196 

97326 

55235 

97044 

6 

55 

51080 

97589 

53231 

97322 

55068 

97039 

5 

56 

51117 

97584 

53266 

97317 

55301 

97035 

4 

57 

51154 

97580 

53301 

97312 

55334 

97030 

3 

58 

51191 

97576 

53336 

97308 

55367 

97025 

2 

59 

51227 

97571 

53370 

97303 

55400 

97020 

1 

60 

51264 

97567 

53405 

9729!) 

55433 

97015 

0 

t 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

71° 

70° 

69° 

TABLE  8.— LOGARITHMIC   SINES  AND  COSINES.    183 


/ 

21° 

22° 

23 

o 

t 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.55433 

9.97015 

9.57358 

9.96717 

9.59188 

9.96403 

60 

1 

55466 

97010 

57389 

96711 

59218 

96397 

59 

2 

55499 

97005 

57420 

96706 

59247 

96392 

58 

3 

5553vJ 

97001 

57451 

96701 

59277 

963^7 

57 

4 

55564 

96996 

57482 

96696 

59307 

96381 

56 

5 

55597 

96991 

57514 

96691 

59336 

96376 

55 

6 

55630 

96986 

57545 

96686 

59366 

96370 

54 

7 

55663 

96981 

57576 

96681 

59396 

96365 

53 

8 

55695 

96976 

57607 

96676 

59425 

96360 

52 

9 

55728 

96971 

57638 

96670 

59455 

96354 

51 

10 

9.55761 

9.96966 

9.57669 

9.96665 

9.59484 

9.96349 

50 

11 

55793 

969(52 

57700 

96C60 

59514 

96343 

49 

12 

55S26 

96957 

57731 

96655 

59543 

96338 

48 

13 

55858 

96952 

57762 

96650 

59573 

96333 

47 

14 

55891 

96947 

57793 

96645 

59602 

96327 

46 

15 

55923 

96942 

57824 

96640 

59632 

96322 

45 

16 

55956 

96937 

57855 

96634 

59661 

96316 

44 

17 

55988 

96932 

57885 

96629 

59690 

96311 

43 

18 

56021 

96927 

57916 

96624 

59720 

96305 

42 

19 

56053 

90922 

57947 

96619 

59749 

96300 

41 

20 

9.56085 

9.96917 

9.57978 

9.96614 

9.59778 

9.96294 

40 

21 

56118 

96912 

5800.8 

96608 

59808 

96289 

39 

22 

56150 

96907 

58039 

96603 

59837 

96284 

38 

23 

56182 

96903 

58070 

96598 

59866 

96278 

37 

24 

56215 

96898 

58101 

96593 

59895 

96273 

36 

25 

56-J47 

96893 

58131 

96588 

59924 

96267 

35 

26 

56279 

96888 

58162 

96582 

59954 

96262 

34 

27 

56311 

96883 

58192 

96577 

59983 

96256 

33 

28 

56343 

96878 

58223 

96572 

60012 

96251 

32 

29 

56375 

96873 

58253 

96567 

60041 

96245 

31 

30 

9.56408 

9.96868 

9.58284 

9.96562 

9.60070 

9.96240 

30 

31 

56440 

96863 

58314 

96556 

60099 

96234 

29 

32 

56472 

96858 

58345 

96551 

60128 

96229 

28 

33 

56504 

96853 

58375 

96546 

60157 

962^3 

27 

34 

56536 

96848 

58406 

96541 

60186 

90218 

26 

35 

56568 

96843 

58436 

96535 

60215 

96212 

25 

36 

56599 

96838 

58467 

96530 

60244 

96207 

24 

37 

56631 

96833 

58497 

965^5 

60273 

96201 

23 

38 

56663 

96828 

58527 

96520 

60302 

96196 

22 

39 

56695 

96823 

58557 

96514 

60331 

96190 

21 

40 

9.56727 

9.96818 

9.58588 

9.96509 

9.60359 

9.96185 

20 

41 

56759 

90813 

58618 

96504 

60^88 

96179 

19 

42 

56790 

96808 

58648 

96498 

60417 

96174 

18 

43 

668*2 

96803 

58678 

96493 

60446 

96168 

17 

44 

56854 

96798 

58709 

96488 

60474 

96162 

16 

45 

56886 

96793 

58739 

96483 

60503 

i!6157 

15 

46 

56917 

96788 

58769 

96477 

60532 

96151 

14 

47 

56949 

96783 

58799 

96472 

60561 

96146 

13 

48 

56980 

96778 

58829 

96467 

60589 

96140 

12 

49 

57012 

90772 

58859 

96461 

60618 

96135 

11 

50 

9.57044 

9.96767 

9.58889 

9.96456 

9.60646 

9.96129 

10 

51 

57075 

96702 

58919 

96451 

60675 

96123 

9 

52 

57107 

96757 

58949 

96445 

60704 

96118 

8 

53 

57138 

96752 

58979 

96440 

60732 

96112 

7 

54 

57169 

90747 

59009 

96435 

60761 

96107 

6 

55 

57201 

96742 

59039 

96429 

60789 

96101 

5 

56 

96737 

59069 

96424 

60818 

96095 

4 

57 

57264 

96732 

59098 

96419 

60846 

96090 

8 

58 

57295 

96727 

59128 

96413 

60875 

96084 

2 

59 

57326 

96722 

59158 

96408 

60903 

96079 

1 

60 

57358 

96717 

59188 

96403 

60931 

96073 

0 

t 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

i 

68° 

67° 

66° 

184     TABLE  8.— LOGARITHMIC   SINES  AND  COSINES. 


24° 

25° 

\ 

6° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

o 

9.60931 

9.96073 

9.62595 

9.95728 

9.64184 

9.95366 

60 

1 

60960 

96067 

62622 

95722 

64210 

95360 

59 

2 

60988 

96062 

62649 

95716 

64236 

95354 

58 

3 

61016 

96056 

62676 

95710 

64262 

95348 

57 

4 

61045 

96050 

62703 

95704 

64288 

95341 

56 

5 

61073 

96045 

62730 

95698 

64313 

95335 

55 

6 

61101 

96039 

62757 

95692 

64339 

95329 

54 

7 

61129 

96034 

62784 

95686 

64365 

95323 

53 

8 

61158 

96028 

62811 

95680 

64391 

95317 

52 

9 

61186 

96022 

62838 

95674 

64417 

95310 

51 

10 

9.61214 

9.96017 

9.62865 

9.95668 

9.64442 

9.95304 

50 

11 

61242 

96011 

62892 

95663 

64468 

95298 

49 

12 

61270 

96005 

62918 

95657 

64494 

95292 

48 

13 

61298 

96000 

62945 

95651 

64519 

95286 

47 

14 

61326 

95994 

62972 

95645 

64545 

95279 

46 

15 

61354 

95988 

62999 

95639 

64571 

95273 

45 

16 

61382 

95982 

63026 

95633 

64596 

95267 

44 

17 

61411 

95977 

63052 

95627 

64622 

95261 

43 

18 

61438 

95971 

63079 

95621 

64647 

95254 

42 

19 

61466 

95965 

63106 

95615 

64673 

95248 

41 

20 

9.61494 

9.95960 

9.63133 

9.95609 

9.64698 

9.95242 

40 

21 

61522 

95954 

63159 

95603 

64724 

95236 

39 

22 

61550 

95948 

63186 

95597 

64749 

95229 

38 

23 

61578 

95942 

63213 

95591 

64775 

95223 

37 

24 

61606 

95937 

63239 

95585 

64800 

95217 

36 

25 

61634 

95931 

632G6 

95579 

64826 

95211 

35 

26 

61662 

95925 

63292 

95573 

64851 

95204 

34 

27 

61689 

95920 

63319 

95567 

64877 

95198 

33 

28 

61717 

95914 

63345 

95561 

64902 

95192 

32 

29 

61745 

95908 

63372 

95555 

64927 

95185 

31 

30 

9.61773 

9.95902 

9.63398 

9.95549 

9.64953 

9.95179 

30 

31 

61800 

95897 

63425 

95543 

64978 

95173 

29 

32 

618-28 

95891 

63451 

95537 

65003 

95167 

28 

33 

61856 

95885 

63478 

95531 

65029 

95160 

27 

34 

61883 

95879 

C3504 

95525 

65054 

95154 

26 

35 

61911 

95873 

63531 

95519 

65079 

95148 

25 

86 

61939 

95868 

63557 

95513 

65104 

95141 

24 

37 

61966 

95862 

63583 

95507 

65130 

95135 

23 

38 

61994 

95856 

63610 

95500 

65155 

95129 

22 

39 

62021 

95850 

63636 

95494 

65180 

95122 

21 

40 

9.62049 

9.95844 

9.63662 

9.95488 

9.65205 

9.95116 

20 

41 

62076 

95839 

63689 

95482 

65230 

95110 

19 

42 

62104 

95833 

63715 

95476 

65255 

95103 

18 

43 

62131 

95827 

63741 

95470 

65281 

95097 

17 

44 

62159 

95821 

63767 

95464 

65306 

95090 

16 

45 

62186 

95815 

63794 

95458 

65331 

95084 

15 

46 

62214 

95810 

63820 

95452 

65356 

95078 

14 

47 

62241 

95804 

63846 

95446 

65381 

95071 

13 

48 

62268 

95798 

63872 

95440 

65406 

95065 

12 

49 

62296 

95792 

63898 

95431 

65431 

95059 

11 

50 

9.62323 

9.95786 

9.63924 

9.95427 

9.65456 

9.95052 

10 

51 

62350 

95780 

63950 

95421 

65481 

95046 

9 

52 

62377 

95775 

63976 

95415 

65506 

95039 

8 

53 

62405 

95769 

64002 

95409 

65531 

95033 

7 

54 

62432 

95763 

64028 

95403 

65565 

95027 

6 

55 

62459 

95757 

64054 

95397 

65580 

95020 

5 

56 

62486 

95751 

64080 

95391 

65605 

95014 

4 

57 

62513 

95745 

64106 

95384 

65630 

95007 

3 

58 

62541 

95739 

64132 

95378 

65655 

95001 

2 

59 

62568 

95733 

64158 

95372 

65680 

94995 

1 

60 

62595 

95728 

64184 

95366 

65705 

94988 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

66° 

64° 

63° 

TABLE  8.— LOGARITHMIC   SINES  AND  COSINES.  185 


/ 

27° 

28° 

29° 

t 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.65705 

9.94988 

9.67161 

9.94593 

9.68557 

9.94182 

60 

1 

65729 

94982 

67185 

94587 

68580 

94175 

59 

2 

65754 

94975 

67208 

94580 

68603 

94168 

58 

3 

65779 

94969 

67232 

94573 

68625 

94161 

57 

4 

65804 

94962 

67256 

94567 

68648 

94154 

56 

5 

65828 

94956 

07280 

94560 

68671 

94147 

55 

6 

65853 

94949 

67303 

94553 

68694 

94140 

54 

7 

65878 

94943 

67327 

94546 

68716 

94133 

53 

8 

65902 

94936 

67350 

94540 

68739 

94126 

52 

9 

65927 

94930 

67374 

94533 

68762 

94119 

51 

10 

9.65952 

9.94923 

9.67398 

9.94526 

9.68784 

9.94112 

50 

11 

65976 

94917 

67421 

94519 

68807 

94105 

49 

12 

66001 

94911 

67445 

94513 

68829 

94098 

48 

18 

66025 

94904 

67468 

94506 

68852 

94090 

47 

14 

66050 

94898 

67492 

94499 

68875 

94083 

46 

15 

66075 

94891 

67515 

94492 

68897 

94076 

45 

16 

66099 

94885 

67539 

94485 

68920 

94069 

44 

17 

66124 

94878 

67562 

94479 

68942 

94062 

43 

18 

66148 

94871 

67586 

94472 

68965 

94055 

42 

19 

66173 

94805 

67609 

94465 

68987 

94048 

41 

20 

9.66197 

9.94858 

9.67633 

9.94458 

9.69010 

9.94041 

40 

21 

66221 

94852 

67656 

94451 

69032 

94034 

39 

22 

66246 

94845 

67680 

94445 

69055 

94027 

38 

23 

66270 

94839 

67703 

94438 

69077 

94020 

37 

24 

66295 

9J832 

67726 

94431 

69100 

94012 

36 

25 

66319 

94826 

67750 

94424 

69122 

94005 

35 

26 

66343 

94819 

67773 

91417 

69144 

93998 

34 

27 

66368 

94813 

67796 

94410 

69167 

93991 

33 

28 

66392 

94806 

67820 

94404 

69189 

S3984 

82 

29 

66416 

94799 

67843 

94397 

69212 

93977 

31 

30 

9.66441 

9.94793 

9.67866 

9.94390 

9.69234 

9.93970 

30 

31 

66465 

94786 

67890 

94883 

69256 

93963 

29 

32 

66489 

94780 

67913 

94376 

69279 

93955 

28 

33 

66513 

94773 

67936 

94369 

69301 

93948 

27 

34 

66537 

94767 

67959 

94362 

69323 

93941 

26 

*  35 

66562 

94760 

67982 

94355 

69345 

93934 

25 

36 

665^6 

94753 

68006 

94349 

69368 

93927 

24 

37 

66610 

94747 

68029 

94342 

69390 

93920 

23 

38 

66634 

94740 

68052 

94335 

69412 

93912 

22 

39 

66G58 

94734 

68075 

94328 

69434 

93905 

21 

40 

9.66682 

9.94727 

9.68098 

9.94321 

9.69456 

9.93898 

20 

41 

66706 

94720 

68121 

94314 

69479 

93891 

19 

42 

66731 

94714 

68144 

94307 

69501 

93884 

18 

43 

66755 

94707 

68167 

94300 

69523 

93876 

17 

44 

66779 

94700 

68190 

94293 

69545 

93869 

16 

45 

66803 

94694 

68213 

94286 

69567 

93862 

15 

46 

66827 

9468? 

68237 

94279 

69589 

93855 

14 

47 

66851 

94680 

68260 

94273 

69611 

93847 

13 

48 

66875 

94674 

68283 

94266 

69638 

93840 

12 

49 

66899 

94667 

68305 

94259 

69655 

93833 

11 

50 

9.66922 

9.94660 

9.68328 

9.94252 

9.69677 

9.93826 

10 

51 

66946 

94654 

68351 

94245 

69699 

93819 

9 

52 

66970 

94647 

68374 

94238 

93811 

8 

53 

66994 

94640 

68397 

94231 

69743 

93804 

7 

54 

67018 

94634 

68420 

94224 

69765 

93797 

6 

55 

67042 

94627 

68443 

94217 

69787 

93789 

5 

66 

67066 

94620 

68466 

94210 

69809 

93782 

4 

57 

67090 

94614 

68489 

94203 

69831 

93775 

3 

58 

67113 

94607 

68512 

94196 

69853 

93768 

2 

59 

67137 

94GOO 

68534 

94189 

69875 

93760 

1 

60 

67161 

94593 

68557 

94182 

69897 

93753 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

62° 

61 

o 

60° 

186     TABLE  8.—  LOGARITHMIC   SINES  AND  COSINES. 


$ 

80° 

81° 

8 

!2° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.69897 

9.93753 

9.71184 

9.93307 

9.72421 

9.92842 

60 

1 

69919 

93746 

71205 

93299 

72441 

92834 

59 

2 

69941 

93738 

71226 

93291 

72461 

92826 

58 

3 

69963 

93731 

71247 

93284 

72482 

92818 

57 

4 

69984 

93724 

71268 

93276 

72502 

92810 

50 

5 

70006 

93717 

71289 

93269 

72528 

92803 

55 

6 

70028 

93709 

71310 

93261 

72542 

92795 

54 

7 

70050 

93702 

71331 

93253 

7256-3 

92787 

53 

8 

70072 

93595 

71352 

93246 

72582 

92779 

52 

9 

70093 

93687 

71373 

93238 

72602 

92771 

51 

10 

9.70115 

9.93680 

9.71393 

9.93230 

9  726°2 

9.92703 

50 

11 

70137 

93673 

71414 

932,3 

72048 

92755 

49 

12 

70159 

93665 

71435 

93215 

72063 

92747 

48 

13 

70180 

93658 

71456 

93207 

7208  J 

92739 

47 

14 

70202 

93650 

71477 

93200 

72703 

92731 

46 

15 

70224 

93643 

71498 

93192 

72723 

92723 

45 

16 

70245 

93636 

71519 

93184 

72743 

92715 

44 

17 

70267 

93628 

71539 

93177 

72763 

92707 

43 

18 

70288 

936'>1 

71560 

93169 

72783 

92099 

42 

19 

70310 

93614 

71581 

93161 

72803 

92691 

41 

20 

9.70332 

9.93606 

9.71602 

9.93154 

9.72828 

9.92683 

40 

21 

70353 

93599 

71622 

93146 

72843 

92675 

39 

22 

70375 

93591 

71643 

93138 

72863 

92667 

38 

23 

70396 

93584 

71664 

93131 

72883 

92659 

37 

24 

70418 

93577 

71685 

93123 

72902 

92051 

36 

25 

70439 

93569 

71705 

93115 

72922 

92043 

35 

26 

70461 

93562 

71726 

93108 

72942 

92635 

34 

27 

70482 

93554 

71747 

93100 

72962 

92027 

33 

28 

70504 

93547 

71767 

93092 

72982 

92019 

32 

29 

70525 

93539 

71788 

93084 

73002 

92011 

31 

80 

9.70547 

9.93532 

9.71809 

9.93077 

9.73022 

9.92603 

30 

81 

70508 

93525 

71829 

93069 

73041 

92595 

29 

32 

70590 

93517 

71850 

93061 

73061 

92587 

28 

33 

70611 

93510 

71870 

93053 

73081 

92579 

27 

34 

70633 

93502 

71891 

93046 

73101 

92571 

26 

35 

70654 

93495 

71911 

93038 

73121 

92563 

25 

36 

70675 

93487 

71932 

93030 

73140 

92555 

24 

37 

70697 

93480 

7195-J 

93022 

73160 

92546 

23 

38 

70718 

93472 

71973 

93014 

73180 

92538 

22 

39 

70739 

93465 

71994 

93007 

73200 

92530 

21 

40 

9.70761 

9.93457 

9.72014 

9.92999 

9.73219 

9.92522 

20 

41 

70782 

93450 

72034 

92991 

73239 

92514 

19 

42 

70803 

93442 

72055 

92983 

73259 

92500 

18 

43 

70824 

934:35 

72075 

92976 

73278 

92498 

17 

44 

70846 

93427 

72096 

92968 

73298 

92490 

16 

45 

70867 

93420 

72116 

92960 

73318 

92482 

15 

46 

70888 

93412 

72137 

92952 

73337 

92473 

14 

47 

70909 

93405 

72157 

92944 

73357 

92465 

13 

48 

70931 

93397 

72177 

92936 

73377 

92457 

12 

49 

70952 

93390 

72198 

92929 

73396 

92449 

11 

50 

9.70973 

9.93382 

9.72218 

9.92921 

9.73416 

9.92441 

10 

51 

70994 

93375 

72238 

92913 

73435 

92433 

9 

62 

71015 

93367 

72259 

92905 

73455 

92425 

8 

53 

71036 

93360 

72279 

92897 

73474 

92416 

7 

54 

71058 

93352 

72299 

92889 

73494 

92408 

6 

55 

71079 

93344 

72320 

92881 

73513 

92400 

5 

56 

71100 

93337 

72340 

92874 

73533 

92392 

4 

57 

71121 

93329 

72360 

92866 

73552 

92384 

3 

58 

71142 

93322 

72881 

92858 

73572 

92376 

2 

59 

71163 

93314 

72401 

92850 

73591 

92367 

1 

60 

71184 

93307 

72421 

92842 

73611 

92359 

0 

/ 

Cosine 

Sine 

Cosine    Sine 

Cosine 

Sine 

/ 

69° 

58° 

67° 

TABLE  8.— LOGARITHMIC   SINES  AND  COSINES.   187 


88 

84° 

35 

0 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.73611 

9.92359 

9.74756 

9.91857 

9.75859 

9.91336 

60 

1 

73630 

92351 

74775 

91849 

75877 

91328 

59 

2 

73650 

92343 

74794 

91840 

75895 

91319 

58 

3 

73669 

92335 

74812 

91832 

75913 

91310 

57 

4 

73689 

92326 

74&31 

91823 

75931 

91301 

56 

5 

73708 

92318 

74850 

91815 

75949 

91292 

55 

6 

73727 

92310 

74868 

91806 

75967 

91288 

54 

7 

73747 

92302 

74887 

91798 

75985 

91274 

53 

8 

73766 

92293 

74906 

91789 

76003 

91266 

52 

9 

73785 

92285 

74924 

91781 

76021 

91257 

51 

10 

9.73805 

9.92277 

9.74943 

9.91772 

9.76039 

9.91248 

50 

11 

73824 

92269 

74961 

91763  . 

.  76057 

91239 

49 

12 

73843 

922GO 

74980 

91755 

76075 

91230 

48 

13 

73863 

92252 

74999 

91746 

76093 

91221 

47 

14 

73SS2 

92244 

75017 

91738 

76111 

91212 

46 

15 

73901 

92235 

75036 

91729 

76129 

91203 

45 

16 

73921 

92227 

75054 

91720 

76146 

91194 

44 

17 

73940 

92219 

75073 

91712 

76164 

91185 

43 

18 

73959 

92211 

75091 

91703 

76182 

91176 

42 

19 

73978 

92202 

75110 

91695 

76200 

91167 

41 

20 

9.73997 

9.92194 

9.75128 

9.  91  f  86 

9.76218 

9.91158 

40 

21 

74017 

92186 

75147 

91677 

76236 

91149 

39 

22 

74036 

92177 

75165 

91669 

76253 

91141 

38 

28 

74055 

92169 

75184 

91660 

76271 

91132 

37 

24 

74074 

92161 

75202 

91651 

76289 

91123 

36 

25 

74093 

92152 

75221 

91643 

76307 

91114 

35 

26 

74113 

92144 

75239 

916^4 

76324 

91105 

34 

27 

74132 

92136 

75258 

91625 

76342 

91096 

33 

28 

74151 

92127 

75276 

91617 

76360 

91087 

32 

29 

74170 

92119 

75294 

91608 

76378 

91078 

31 

30 

9.74189 

9.92111 

9.75313 

9.91599 

9.76395 

9.91069 

30 

31 

74208 

92102 

75331 

91591 

76413 

91060 

29 

32 

74227 

92094 

75350 

91582 

76431 

91051 

28 

33 

74246 

92086 

75368 

91573 

76448 

91042 

27 

34, 

r,4265 

92077 

75386 

91565 

76466 

91033 

26 

35 

74284 

92069 

75405 

91556 

76484 

91023 

25 

36 

74303 

92060 

75423 

91547 

76501 

91014 

24 

sr 

74322 

92052 

75441 

91538 

76519 

91005 

23 

38 

74341 

92044 

75459 

91530 

76537 

90996 

22 

39 

74360 

92035 

75478 

91521 

76554 

90987 

21 

40 

9.74379 

9.92027 

9.75496 

9.91512 

9.76572 

9.90978 

20 

41 

74398 

92018 

75514 

91504 

76590 

90969 

19 

42 

74417 

92010 

75533 

91495 

76607 

90960 

18 

43 

74436 

92002 

75551 

91486 

76625 

90951 

17 

44 

74455 

91993 

75569 

91477 

76642 

90942 

16 

45 

74474 

91985 

75587 

91469 

76660 

90933 

15 

46 

74493 

91976 

75605 

91460 

76677 

90924 

14 

47 

74512 

91968 

75624 

91451 

76695 

90915 

13 

48 

74531 

91959 

75642 

91442 

76712 

90906 

12 

49 

74549 

91951 

75660 

91433 

76730 

90696 

11 

50 

9.74568 

9.91942 

9.75678 

9.91425 

9.76747 

9.90887 

10 

51 

74587 

91934 

75696 

91416 

76765 

90878 

9 

52 

74606 

91925 

75714 

91407 

76782 

90869 

8 

53 

74625 

91917 

75733 

91398 

76800 

90860 

7 

54 

74644 

91908 

75751 

91389 

76817 

90851 

6 

55 

74662 

91900 

75769 

91381 

76835 

90842 

5 

56 

74681 

91891 

75787 

91372 

76852 

90832 

4 

57 

74700 

91883 

75805 

91363 

76870 

90823 

3 

58 

74719 

91874 

75823 

91354 

76887 

90814 

2 

59 

74737 

91866 

75841 

91345 

76904 

90805 

1 

60 

74756 

91857 

75859 

91336 

76922 

90796 

0 

t 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

f 

66° 

66° 

64° 

188     TABLE  8.— LOGARITHMIC   SINES  AND  COSINES. 


t 

86° 

37° 

38° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.76922 

9.90796 

9.77946 

9.90235 

9.78934 

9.89653 

60 

1 

76939 

90787 

77963 

90225 

78950 

89643 

59 

2 

76957 

90777 

77980 

90216 

78967 

89633 

58 

3 

76974 

90768 

77997 

90206 

78983 

89624 

57 

4 

76991 

90759 

78013 

90197 

78999 

89614 

56 

5 

77009 

90750 

78030 

90187 

79015 

89604 

55 

6 

77026 

90741 

78047 

90178 

79031 

89594 

54 

7 

77043 

90731 

78063 

90168 

79047 

89584 

53 

8 

77061 

90722 

78080 

90159 

79063 

89574 

52 

9 

77078 

90713 

78097 

90149 

79079 

89564 

51 

10 

9.77095 

9.90704 

9.78113 

9.90139 

9.79095 

9.89554 

50 

11 

77112 

90694 

78130 

90130 

79111 

89544 

49 

12 

77130 

90685 

78147 

90120 

79128 

89534 

48 

13 

77147 

90676 

78163 

90111 

79144 

89524 

47 

14 

77164 

90667 

78180 

90101 

79160 

89514 

46 

15 

77181 

90657 

78197 

90091 

79176 

89504 

45 

16 

77199 

90648 

78213 

90082 

79192 

89495 

44 

17 

77216 

90639 

78230 

90072 

79208 

89485 

43 

18 

77233 

90630 

78246 

90063 

79224 

89475 

42 

19 

77250 

90620 

78263 

90053 

79240 

89465 

41 

20 

9.77268 

9.90611 

9.73280 

9.90043 

9.79256 

9.89455 

40 

21 

77285 

90602 

78296 

90034 

79272 

89445 

39 

22 

77302 

90592 

78313 

,  90024 

79288 

89435 

38 

23 

77319 

90583 

78329 

90014 

79304 

89425 

37 

24 

77336 

90574 

78346 

90005 

19319 

89415 

36 

25 

77353 

90565 

7836-2 

89995 

79335 

89405 

35 

26 

77370 

90555 

78379 

89985 

79351 

89395 

34 

27 

7738  r 

90546 

78395 

89976 

79367 

89385 

33 

28 

77405 

90537 

78412 

89966 

79383 

89375 

32 

29 

77422 

905:27 

78428 

89956 

Y9399 

89364 

31 

30 

9.77439 

9.90518 

9.78445 

9.89947 

9.79415 

9.89354 

30 

31 

77456 

90509 

78461 

89937 

79431 

89344 

29 

38 

77473 

90499 

78478 

89927 

79447 

89334 

28 

33 

77490 

90490 

78494 

89918 

79463 

89824 

27 

34 

77507 

90480 

78510 

89908 

79478 

89314 

26 

35 

77524 

90471 

785-27 

89898 

79494 

89304 

25 

36 

77541 

90462 

78543 

89888 

79510 

89294 

24 

37 

77558 

90452 

78560 

89879 

79526 

89284 

23 

38 

77575 

90443 

78576 

89869 

79542 

89274 

22 

39 

77592 

90434 

7859;! 

89859 

79558 

89264 

21 

40 

9.77609 

9.90424 

9.78609 

9.89849 

9.79573 

9.89254 

20 

41 

77626 

90415 

78625 

89840 

79589 

89244 

19 

42 

"7643 

90405 

7864-2 

89830 

79605 

89233 

18 

48 

"7060 

90396 

78658 

89820 

78621 

89223 

17 

44 

"7677 

90386 

78674 

89810 

79636 

89213 

16 

45 

"7694 

90377 

78691 

89801 

79652 

89203 

15 

46 

"7711 

9036S 

78707 

89791 

79668 

89193 

14 

47 

""7728 

90358 

78723 

89781 

79684 

89183 

13 

48 

7744 

90349 

78739 

89771 

79699 

89173 

12 

49 

"7761 

90339 

78756 

89761 

79715 

89162 

11 

50 

9.77778 

9.90330 

9.78772 

9.89752 

9.79731 

9.89152 

10 

51 

77795 

90320 

78788 

89742 

79746 

89142 

9 

52 

77812 

90311 

78805 

89732 

79762 

89132 

8 

53 

77849 

90301 

78821 

89722 

79778 

89122 

7 

54 

77846 

90292 

78837 

89712 

79793 

89112 

6 

55 

77862 

90282 

78853 

89702 

79809 

89101 

5 

f)6 

77879 

90273 

78869 

89693 

79825 

89091 

4 

57 

77896 

90263 

78886 

89683 

79840 

89081 

3 

58 

77913 

90254 

78902 

89673 

79856 

89071 

2 

59 

77930 

90244 

78918 

89663 

79872 

89060 

1 

GO 

77946 

90235 

78934 

89653 

79887 

89050 

0 

* 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

68° 

62° 

61° 

TABLE  8.— LOGARITHMIC   SINES  AND  COSINES.     189 


/ 

39° 

40° 

41° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.79887 

9.89050 

9.80807 

9.88425 

9.81694 

9.87778 

60 

1 

79903 

89040 

80822 

88415 

81709 

87767 

59 

2 

79918 

89030 

80837 

88404 

81723 

87756 

58 

3 

79934 

89020 

80852 

88394 

81738 

87745 

57 

4 

79950 

89009 

80867 

88383 

81752 

87734 

56 

5 

79965 

88999 

80882 

88372 

81767 

87723 

55 

6 

79981 

88989 

80897 

88362 

81781 

87712 

54 

7 

79996 

88978 

80912 

88351 

81796 

87701 

53 

8 

80012 

88968 

80927 

88340 

81810 

87690 

52 

9 

80027 

88958 

80942 

88330 

81825 

87679 

51 

10 

9.80043 

9.88948 

9.80957 

9.88319  . 

9.81839 

9.87668 

50 

11 

80058 

88937 

80972 

88308 

81854 

87657 

49 

12 

80074 

88927 

80987 

88298 

81868 

87646 

48 

13 

80089 

88917 

81002 

88287 

81882 

87635 

47 

14 

80105 

88906 

81017 

88276 

81897 

87624 

46 

15 

80120 

88896 

81032 

88266 

81911 

87613 

45 

16 

80136 

88886 

81047 

88255 

81926 

87601 

44 

17 

80151 

88875 

81061 

88244 

81940 

87590 

43 

18 

80166 

88865 

81076 

88234 

81955 

87579 

42 

19 

80182 

88855 

81091 

88223 

61969 

87568 

41 

20 

9.80197 

9.88844 

9.81106 

9.88212 

9.81983 

9.87557 

40 

21 

80213 

88834 

81121 

88201 

81998 

87546 

39 

22 

80228 

88824 

81136 

88191 

82012 

87535 

38 

23 

80244 

88813 

81151 

88180 

8-J026 

87524 

37 

24 

80259 

8S803 

81166 

88169 

82041 

87513 

36 

& 

80274 

88793 

81180 

88158 

82055 

87501 

35 

26 

80290 

88782 

81195 

88148 

82069 

87490 

34 

27 

80305 

88772 

81210 

88137 

82084 

87479 

33 

28 

80320 

88761 

81225 

88*26 

82098 

87468 

82 

29 

80336 

88751 

81240 

88115 

82112 

87457 

31 

30 

9.80351 

9.88741 

9.81254 

9.88105 

9.82126 

9.87446 

30 

31 

80366 

88730 

81269 

88094 

82141 

87434 

29 

32 

80382 

88720 

81284 

88083 

82155 

87423 

£8 

m 

80397 

88709 

81299 

88072 

82169 

87412 

27 

34 

80412 

88699 

81314 

88061 

82184 

87401 

M 

35 

80428 

88688 

81328 

88051 

82198 

87390 

25 

36 

80443 

88678 

81343 

88040 

82212 

87378 

24 

37 

80458 

88668 

81358 

88029 

82226 

87367 

23 

38 

80473 

88657 

81372 

88018 

82240 

87356 

22 

39 

80489 

88647 

81387 

88007 

82255 

87345 

21 

40 

9.80504 

9.88636 

9.81402 

9.87996 

9.82269 

9.87334 

20 

41 

80519 

88626 

81417 

87985 

82283 

87322 

19 

42 

80534 

88615 

81431 

87975 

82297 

87311 

18 

43 

80550 

88605 

81446 

87964 

82311 

87300 

17 

44 

80565 

88594 

81461 

87953 

82326 

87288 

16 

45 

80580 

88584 

81475 

87942 

82340 

87277 

15 

46 

80595 

8*573 

81490 

87931 

82354 

87266 

14 

47 

80610 

88563 

81505 

87920 

82368 

87255 

13 

48 

80625 

88552 

81519 

87909 

82382 

87243 

12 

49 

80641 

88542 

81534 

87898 

82396 

87232 

11 

50 

9.80656 

9.88531 

9.81549 

9.87887 

9.82410 

9.87221 

10 

51 

80671 

88521 

81563 

87877 

82424 

87209 

9 

52 

80686 

88510 

81578 

87866 

82439 

87198 

8 

53 

80701 

88499 

81592 

87855 

824f3 

87187 

7 

54 

80716 

88489 

81607 

87844 

82467 

87175 

6 

55 

80731 

88478 

81622 

87833 

82481 

87164 

5 

56 

80746 

88468 

81636 

•87822 

62495 

87153 

4 

57 

80762 

88457 

81651 

87811 

82509 

87141 

3 

58 

80777 

88447 

81665 

87800 

82523 

87130 

2 

59 

80792 

88436 

81680 

87789 

82537 

87119 

1 

60 

80807 

88425 

81694 

87778 

82551 

87107 

0 

t 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

50° 

49 

o 

48° 

190     TABLE   8.— LOGARITHMIC   SINES  AND  COSINES. 


42° 

48° 

44° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.82551 

9.8710? 

9.83378 

9.86413 

9.84177 

9.85693 

60 

1 

82565 

87096 

83392 

86401 

84190 

85681 

59 

2 

82579 

87085 

83405 

86889 

84203 

85669 

58 

3 

82593 

87073 

83419 

86377 

84216 

85657 

57 

4 

82607 

87062 

83432 

86366 

84229 

85645 

56 

5 

82621 

87050 

83446 

86354 

84242 

85C32 

55 

6 

82635 

87039 

83459 

86342 

84255 

85620 

54 

7 

82649 

87028 

83473 

80330 

84269 

85608 

53 

8 

82663 

87016 

83486 

86318 

84282 

85596 

52 

9 

82677 

87005 

83500 

86306 

84295 

85583 

51 

10 

9.82691 

9.86993 

9.83513 

9.86295 

9.84308 

9.85571 

50 

11 

82705 

86982 

83527 

86283 

84321 

85559 

49 

12 

82719 

86970 

83540 

86271 

84334 

85547 

48 

13 

82733 

86959 

83554 

86259 

84347 

85534 

47 

14 

82747 

86947 

83567 

86247 

84360 

85522 

46 

15 

sarei 

86936 

83581 

86235 

84373 

85510 

45 

16 

82775 

86924 

83594 

86223 

84385 

85497 

44 

17 

82788 

86913 

83608 

86211 

84398 

85485 

43 

18 

82802 

86902 

83621 

86200 

84411 

85473 

42 

19 

82816 

86890 

83634 

86188 

84424 

85460 

41 

20 

9.82830 

9.86879 

9.83648 

9.86176 

9.84437 

9.85448 

40 

21 

82844 

86867 

83661 

86164 

84450 

85436 

39 

22 

82858 

86855 

83674 

86152 

84463 

85423 

38 

23 

82872 

86844 

83688 

86140 

84476 

85411 

37 

24 

82885 

86832 

83701 

86128 

84489 

85399 

36 

25 

82899 

86821 

83715 

86116 

84502 

85386 

35 

26 

82913 

86809 

83728 

86104 

84515 

85874 

34 

27 

82927 

86798 

83741 

86092 

84528 

85361 

33 

28 

82941 

86786 

83755 

86080 

84540 

85349 

32 

29 

82955 

86775 

83768 

86068 

84553 

85337 

31 

30 

9.82968 

9.86763 

9.83781 

9.86056 

9.84566 

9.85324 

30 

31 

82982 

86752 

83795 

86044 

84579 

85312 

29  ' 

32 

82996 

86740 

83808 

86032 

84592 

85299 

28 

33 

83010 

86728 

83821 

86020 

84605 

85287 

27 

34 

83023 

86717 

83834 

86008 

84618 

85274 

26 

a5 

83037 

86705 

83848 

85996 

84630 

85262 

25 

36 

83051 

86694 

83861 

85984 

84643 

85250 

24 

37 

83065 

86682 

83874 

85972 

84656 

85237 

23 

38 

83078 

86670 

83887 

85960 

84669 

85225 

22 

39 

83092 

86659 

83901 

85948 

84682 

85212 

21 

40 

9.83106 

9.86647 

9.83914 

9.85936 

9.84694 

9.85200 

20 

41 

83120 

86635 

83927 

85924 

84707 

85187 

19 

42 

83133 

86624  ' 

83940 

85912 

84720 

85175  . 

18 

43 

83147 

86612 

83954 

85900 

84733 

85162 

17 

44 

83161 

86600 

83967 

85888 

84745 

85150 

16 

45 

83174 

86589 

83980 

85876 

84758 

85137 

15 

46 

88188 

86577 

83993 

85864 

84771 

85125 

14 

47 

83202 

86565 

84006 

85851 

84784 

85112 

13 

48 

83215 

86554 

84020 

85839 

84796 

85100 

12 

49 

832v!9 

86542 

84033 

85827 

84809 

85087 

11 

50 

9.83242 

9.86530 

9.84046 

9.85815 

9.84822 

9.85074 

10 

51 

83256 

86518 

84059 

85803 

84835 

85062 

9 

52 

a3270 

86507 

84072 

85791 

84847 

85049 

8 

53 

83283 

86495 

84085 

85779 

84860 

85037 

7 

54 

83297 

86483 

84098 

85766 

84873 

85024 

6 

55 

83310 

86472 

84112 

85754 

84885 

85012 

5 

56 

83324 

86460 

84125 

85742 

84898 

84999 

4 

57 

83338 

86448 

84138 

85730 

84911 

84986 

3 

58 

83351 

86436 

84151 

85718 

84923 

84974 

2 

59 

83365 

86425 

84164 

85706 

84936 

84961 

1 

60 

83378 

86413 

84177 

85693 

84949 

84949 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

47° 

46* 

46° 

TABLE  9.— LOG.  TANGENTS  AND   COTANGENTS.    191 


0° 

1° 

2° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

—  00 

00 

8.24192 

11.75808 

8.54308 

11.45692 

60 

1 

6.46373 

13.53627 

24910 

75090 

54669 

45331 

59 

2 

76476 

23524 

25616 

74884 

55027 

44973 

58 

3 

94085 

05915 

26312 

73688 

55382 

44618 

57 

4 

7.06579 

12.93421 

26996 

73004 

55734 

44266 

56 

5 

16270 

83730 

27669 

72331 

56083 

43917 

55 

6 

24188 

75812 

28332 

71668 

56429 

43571 

54 

7 

80882 

69118 

28986 

71014 

56773 

43227 

53 

8 

36682 

63318 

29629 

70371 

57114 

4tt86 

62 

9 

41797 

58203 

30263 

69737 

57452 

42548 

51 

10 

7.46373 

12.53627 

8.30888 

11.69112 

8.  57788 

11.42212 

no 

11 

50512 

49488 

31505 

68495 

58121 

41879  I  49 

12 

54291 

45709 

32112 

67888 

58451 

41549 

48 

13 

57767 

42233 

32711 

67289 

58779 

41221 

47 

14 

60986 

39014 

33302 

66698 

59105 

40895 

46 

15 

63982 

36018 

33886 

66114 

59428 

40572 

45 

16 

66785 

33215 

34461 

65539 

59749 

40251 

44 

1? 

69418 

30582 

35029 

64971 

60068 

39932 

43 

18 

71900 

28100 

35590 

64410 

60384 

39616 

42 

19 

74248 

25752 

36143 

63857 

60698 

89302 

41 

20 

7.76476 

12.23524 

8.36689 

11.63311 

8.610T9 

11.38991 

40 

21 

78595 

21405 

37229 

62771 

61319 

38681 

80 

22 

80615 

19385 

37762 

62238 

61626 

38374 

38 

23 

82546 

17454 

38289 

61711 

61931 

38069 

37 

24 

84394 

15606 

38809 

61191 

62234 

37766 

36 

25 

86167 

13833 

39323 

60677 

62535 

37465 

35 

26 

87871 

12129 

39832 

60168 

62834 

87166 

34 

27 

89510 

10490 

40334 

59666 

63131 

36869 

33 

28 

91089 

08911 

40830 

59170 

63426 

36574 

32 

29 

92613 

07387 

41321 

58679 

63718 

36282 

31 

30 

7.94086 

12.05914 

8.41807 

11.58193 

8.64009 

11.35991 

30 

31 

95510 

04490 

42287 

57713 

64298 

35702 

29 

32 

*  96889 

03111 

42762 

57238 

64585 

35415 

28 

33 

98225 

01775 

43232 

56768 

64870 

35130 

27 

34 

99522 

00478 

43696 

56304 

65154 

34846 

26 

35 

8.00781 

11.99219 

44156 

55844 

65435 

34565 

25 

36 

02004 

97996 

44611 

55389 

65715 

34285 

24 

37 

03194 

96806 

45061 

54939 

65993 

34007 

23 

38 

04353 

95647 

45507 

54493 

66269 

33731 

22 

39 

05481 

94519 

45948 

54052 

66543 

33457 

21 

40 

6.06581 

11.93419 

8.46385 

11.53615 

8.66816 

11.38184 

20 

41 

07653 

92347 

46817 

53183 

67087 

82913 

19 

42 

08700 

91300 

47245 

52755 

67356 

32644 

18 

43 

09722 

90278 

47669 

52331 

67624 

32376 

17 

44 

10720 

89280 

48089 

51911 

67890 

32110 

16 

45 

11696 

88304 

48505 

51495 

68154 

31846 

15 

46 

12651 

87349 

48917 

51083 

68417 

31583 

14 

47 

13585 

86415 

49325 

50675 

68678 

31322 

13 

48 

14500 

85500 

49729 

50271 

68938 

31062 

12 

49 

15395 

84605 

50130 

49870 

69196 

30804 

11 

50 

8.16273 

11.83727 

8.50527 

11.49473 

8.69453 

11.30547 

10 

51 

17133 

82867 

50920 

49080 

69708 

20292 

9 

52 

17976 

82024 

51310 

48690 

69962 

30038 

8 

53 

18804 

81196 

51696 

48304 

70214 

29786 

7 

54 

19616 

80384 

52079 

47921 

70465 

29535 

6 

55 

20413 

79587 

52459 

47541 

70714 

29286 

5 

56 

21195 

78805 

52835 

47165 

70962 

29038 

4 

57 

21964 

78036 

53208 

46792 

71208 

28792 

8 

58 

22720 

77280 

53578 

46422 

71453 

2F547 

2 

59 

23462 

76538 

53945 

46055 

71697 

28303 

1 

60 

24192 

75808 

54308 

45692 

71940 

28060 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

89° 

88° 

87° 

192     TABLE  9.— LOG.  TANGENTS   AND   COTANGENTS 


8° 

4° 

5° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

8.71940 

11.28060 

8.84464 

11.15536 

8.94195 

11.05805 

60 

1 

72181 

27819 

84646 

15354 

94340 

05660 

59 

2 

72420 

27580 

84826 

15174 

94485 

05515 

58 

3 

72659 

27341 

85006 

14994 

94630 

05370 

57 

4 

72896 

27104 

85185 

14815 

94773 

05227 

56 

5 

73132 

26868 

85363 

14637 

94917 

05083 

55 

6 

73366 

26634 

85540 

14460 

95060 

04940 

54 

73600 

26400 

85717 

14283 

95202 

04798 

53 

8 

73832 

2616S 

85893 

14107 

95344 

04656 

52 

9 

74063 

25937 

86069 

13931 

95486 

04514 

51 

10 

8.74292 

11.25708 

8.86243 

11.13757 

8.95627 

11.04373 

50 

11 

74521 

25479 

86417 

13583 

95767 

04233 

49 

12 

74748 

25252 

86591 

13409 

95908 

04092 

48 

18 

74974 

25028 

86763 

13237 

96047 

03953 

47 

14 

75199 

24801 

86935 

13065 

96187 

03813 

46 

15 

75423 

24577 

87106 

12894 

96325 

03675 

45 

16 

75645 

24355 

87277 

12723 

96464 

03536 

44 

17 

75867 

24133 

87447 

12553 

96602 

03398 

43 

18 

76087 

JJ3913 

87616 

12384 

96739 

03261 

42 

19 

76306 

23694 

87785 

12215 

96877 

02123 

41 

20 

8.76525 

11.23475 

8.87953 

11.12047 

8.97013 

11.02987 

40 

21 

76742 

23258 

88120 

11880 

97150 

02850 

39 

22 

76958 

23042 

88287 

11713 

97285 

02715 

38 

23 

77173 

22827 

88453 

11547 

97421 

02579 

37 

24 

77387 

22613 

88618 

11382 

97556 

02444 

36 

25 

77600 

22400 

88783 

11217 

97691 

02309 

35 

26 

77811 

22189 

88948 

11052 

97825 

02175 

34 

27 

78022 

21978 

89111 

10889 

97959 

02041 

33 

28 

78232 

21768 

89274 

10726 

98092 

01908 

3* 

29 

78441 

21559 

89437 

10563 

98225 

01775 

31 

30 

8.78649 

11.21351 

8.89598 

11.10402 

8.98358 

11.01642 

30 

31 

78855 

21145 

89760 

10240 

98490 

01510 

29 

32 

79061 

20939 

89920 

10080 

98622 

01378 

28 

83 

79266 

20734 

90080 

09920 

98753 

01247 

27 

34 

79470 

20530 

90240 

09760 

98884 

01116 

26 

35 

79673 

20327 

90399 

09601 

99015 

00985 

25 

36 

79875 

20125 

90557 

09443 

99145 

00855 

24 

37 

80076 

19924 

90715 

09285 

99275 

00725 

23 

38 

80277 

19723 

90872 

09128 

99405 

00595 

22 

39 

80476 

19524 

91029 

08971 

99534 

00466 

21 

40 

8.80674 

11.19326 

8.91185 

11.08815 

8.99662 

11.00338 

20 

41 

80872 

19128 

91340 

08660 

99791 

00209 

19 

42 

81068 

18932 

91495 

08505 

99919 

00081 

18 

43 

81264 

18736 

91650 

08350 

9.00046 

10.99954 

17 

44 

81459 

18541 

91803 

08197 

00174 

99826 

16 

45 

81653 

18347 

91957 

08043 

00301 

99699 

15 

46 

81846 

18154 

92110 

07890 

00427 

99573 

14 

47 

82038 

17962 

92262 

07738 

00553 

99447 

13 

48 

82230 

17770 

92414 

07586 

00679 

99821 

12 

49 

82420 

17580 

92565 

07435 

00805 

99195 

11 

50 

8.82610 

11.17390 

8.92716 

11.07284 

9.00930 

10.99070 

10 

51 

82799 

17201 

92866 

07134 

01055 

98945 

9 

52 

82987 

17013 

93016 

06984 

01179 

98821 

8 

53 

83175 

16825 

93165 

068&5 

01303 

98697 

7 

54 

83361 

16639 

93313 

06687 

01427 

98573 

6 

55 

83547 

16453 

93462 

06538 

01550 

98450 

5 

56 

83732 

16268 

93609 

06391 

01673 

98327 

4 

57 

83916 

16084 

93756 

06244 

01796 

98204 

3 

58 

84100 

15900 

93903 

06097 

01918 

98082 

2 

59 

84282 

15718 

94049 

05951 

02040 

97960 

1 

60 

84464 

15536 

94195 

05805 

02162 

97838 

0 

, 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

t 

86' 

85° 

84« 

TABLE  9.— LOG.  TANGENTS   AND  COTANGENTS.   193 


/ 

6 

0 

7° 

8° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.02162 

10.97838 

9.08914 

10.91086 

9.14780 

10.86220 

60 

1 

02283 

97717 

09019 

90981 

14872 

85128 

59 

2 

02404 

97596 

09123 

90877 

14963 

85037 

58 

3 

02525 

97475 

09227 

90773 

15054 

84946 

57 

4 

02645 

97355 

09330 

90670 

15145 

84855 

56 

5 

02766 

97234 

09434 

90566 

15236 

84764 

55 

6 

02885 

97115 

09537 

90463 

15327 

84673 

54 

7 

C3005 

96995 

09640 

90360 

15417 

84583 

53 

8 

03124 

96876 

09742 

90258 

15508 

84492 

52 

9 

03242 

96758 

09845 

90155 

15598 

84402 

51 

10 

9.03361 

10.96639 

9.09947 

10.90053 

9.15688 

10.84312 

50 

11 

03479 

96521 

10049 

89951 

15777 

84228 

49 

12 

03597 

96403 

10150 

89850 

15867 

84133 

48 

13 

03714 

96286 

10252 

89748 

15956 

84044 

47 

14 

03832 

96168 

10353 

89647 

16046 

83954 

46 

15 

03948 

96052 

10454 

89546 

16135 

83865 

45 

16 

04065 

95935 

10555 

89445 

16224 

83776 

44 

17 

04181 

95819 

10656 

89344 

16312 

83688 

43 

18 

04297 

95703 

10756 

89244 

16401 

83599 

42 

19 

04413 

95587 

10856 

89144 

16489 

83511 

41 

20 

9.045C8 

10.95472 

9.10956 

10.89044 

9.16577 

10.83423 

40 

21 

04643 

95357 

11056 

88944 

16665 

83335 

39 

22 

04758 

95242 

11155 

88845 

16753 

83247 

38 

23 

04873 

95127 

11254 

88746 

16841 

83159 

87 

24 

04987 

95018 

11353 

88647 

16928 

83072 

86 

25 

05101 

94899 

11452 

88548 

17016 

P2984 

35 

26 

05214 

94786 

11551 

88449 

17103 

82897 

34 

27 

05378 

94672 

11649 

88351 

17190 

82810 

33 

28 

05441 

94559 

11747 

882f.3 

17277 

82723 

82 

29 

^  05553 

94447 

11845 

88155 

17363 

82637 

31 

30 

9.05666 

10.94334 

9.11943 

10.88057 

9.17450 

10.82550 

30 

31 

05778 

94222 

12040 

87960 

17536 

82464 

29 

32 

05890 

94110 

12138 

87862 

17622 

82378 

28 

33 

06002 

93998 

12235 

87765 

17708 

82292 

27 

34 

06113 

93887 

12332 

87668 

17794 

82206 

26 

35 

06224 

93776 

12428 

87572 

17880 

82120 

25 

36 

06335 

93665 

12525 

87475 

17965 

82035 

24 

37 

06445 

93555 

12621 

87379 

18051 

81949 

23 

38 

06556 

93444 

12717 

87283 

18136 

81864 

22 

39 

06666 

93334 

12813 

87187 

18221 

81779 

21 

40 

9.06775 

10.93225 

9.12909 

10.87091 

9.18306 

10.81694 

20 

41 

06885 

93115 

13004 

86996 

18391 

81609 

19 

42 

06994 

93006 

13099 

86901 

18475 

81525 

18 

43 

07103 

92897 

13194 

86806 

18560 

81440 

17 

44 

07211 

92789 

13289 

86711 

18644 

81856 

16 

45 

07820 

92680 

13384 

86616 

18728 

81272 

15 

46 

07428 

92572 

13478 

86522 

18812 

81188 

14 

47 

\536 

92464 

13573 

86427 

18896 

81104 

13 

48 

07643 

92357 

13667 

86338 

18979 

81021 

K 

49 

07751 

92249 

13761 

86239 

19063 

80937 

11 

50 

9.07858 

10.92142 

9.13854 

10.86146 

9.19146 

10.80854 

10 

51 

07964 

92036 

13948 

86052 

19229 

80771 

9 

52 

08071 

91929 

14041 

85959 

19312 

80688 

8 

53 

08177 

91823 

14134 

85866 

19395 

80605 

1 

54 

08288 

91717 

14227 

85773 

19478 

80522 

6 

55 

08389 

91611 

14320 

85680 

19561 

80439 

5 

56 

08495 

91505 

14412 

85588 

1964:3 

80357 

4 

57 

08600 

91400 

14504 

85496 

19725 

80275 

3 

58 

08705 

91295 

14597 

85403 

19807 

80193 

2 

59 

08810 

91190 

14688 

85312 

19889 

80111 

1 

60     08914 

91086 

14780 

85220 

19971 

800-^9 

0 

, 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

i 

83° 

82» 

81° 

194       TABLE  9.— LOG.  TANGENTS   AND   COTANGENTS. 


t 

9° 

10° 

11° 

/  1 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.19971 

10.800-29 

9.24632 

10.75368 

9.28865 

10.71135 

60 

1 

20053 

79947 

24706 

75294 

28933 

71067 

59 

2 

20134 

79S66 

24779 

75221 

29000 

71000 

58 

3 

20-216 

79784 

24853 

75147 

29067 

70933 

57 

4 

20297 

79703 

24926 

75074 

29134 

70866 

56 

5 

20378 

79622 

25000 

75000 

29201 

70799 

55 

6 

20459 

79541 

25073 

74927 

29268 

70732 

54 

7 

20540 

79460 

25146 

74854 

29335 

70605 

53 

8 

20621 

79379 

25219 

74781 

29402 

70598 

52 

9 

20701 

79299 

25292 

74708 

29468 

70532 

51 

10 

9.20782 

10.79218 

9.25365 

10.74635 

9.29535 

10.70465 

50 

11 

20862 

79138 

25437 

74563 

29601 

70399 

49 

If 

20942 

79058 

25510 

74490 

29668 

70332 

48 

13 

21022 

78978 

25582 

74418 

29734 

70266 

47 

14 

21102 

78898 

25655 

74345 

29800 

70200 

46 

15 

21182 

78818 

25727 

74273 

29866 

70134 

45 

16 

21261 

78739 

25799 

74201 

29932 

70068 

44 

17 

21341 

78659 

25871 

74129 

29998 

70002 

43 

18 

21420 

78580 

25943 

74057 

30064 

69936 

42 

19 

21499 

78501 

26015 

73985 

30130 

69870 

41 

20 

9.21578 

10.78422 

9.26086 

10.73914 

9.30195 

10.69805 

40 

21 

21657 

78343 

26158 

73842 

30261 

69739 

39 

22 

21736 

78264 

26229 

73771 

30326 

69674 

38 

*3 

21814 

78186 

26301 

73699 

30391 

69609 

37 

24 

21893 

78107 

26372 

73628 

30457 

69543 

36 

25 

21971 

78029 

26443 

73557 

30522 

69478 

35 

26 

22049 

77951 

26514 

73486 

30587 

69413 

34 

27 

22127 

77873 

26585 

30652 

69348 

33 

28 

22205 

77795 

26655 

30717 

69283 

32 

29 

22283 

77717 

26726 

30782 

69218 

31 

30 

9.22361 

10.77639 

9.26797 

10.73203 

9.30846 

10.69154 

30 

31 

22438 

7756C 

26867 

73133 

30911 

69089 

29 

33 

22516 

77484 

26937 

73063 

30975 

69025 

28 

33 

225,98 

77407 

27008 

72992 

31040 

68960 

27 

34 

22670 

77330 

27078 

72922 

31104 

68896 

26 

35 

22747 

77253 

27148 

72852 

31168 

68832 

25 

36 

22824 

77176 

27218 

72782 

31233 

68767 

24 

37 

22901 

77099 

27288 

72712 

31297 

68703 

23 

38 

22977 

77023 

27357 

72648 

31361 

68639 

22 

39 

23054 

76946 

27427 

72573 

31425 

68575 

21 

40 

9.23130 

10.76870 

9.27496 

10.72504 

9.31489 

10.68511 

20 

41 

28206 

76794 

27566 

72434 

31552 

68448 

19 

42 

23283 

76717 

27635 

72365 

31616 

68384 

18 

43 

23359 

76641 

27704 

72296 

31679 

68321 

17 

44 

23435 

76565 

27773 

72227 

31743 

68257 

16 

45 

23510 

76490 

27842 

72158 

31806 

68194 

15 

46 

23586 

76414 

27911 

72089 

31870 

68130 

14 

47 

23661 

76339 

27980 

72020 

31933 

68067 

13 

48 

23787 

76263 

28049 

71951 

31996 

68004 

12 

49 

23812 

76188 

28117 

71883 

32059 

67941 

11 

50 

9.23887 

10.76113 

9.28186 

10.71814 

9.32122 

10.67878 

10 

51 

23962 

76038 

28254 

71746 

32185 

67815 

9 

52 

24037 

75963 

28323 

71677 

32248 

67752 

8 

53 

24112 

75888 

28391 

71609 

32311 

67689 

7 

54 

24186 

75814 

28459 

71541 

32373 

67627 

6 

55 

24261 

75739 

28527 

71473 

3243G 

67564 

5 

56 

24335 

75665 

28595 

71405 

32498 

67502 

4 

57 

24410 

75590 

28662 

71338 

32561 

67439 

3 

58 

24484 

75516 

28730 

71270 

32623 

67377 

2 

59 

24558 

75442 

28798 

71202 

32685 

67315 

1 

60 

24632 

75368 

28865 

71135 

32747 

67253 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

80* 

79° 

78° 

TABLE  9.— LOG.  TANGENTS  AND  COTANGENTS.  195 


12° 

18° 

14° 

/ 

Tan 

Cotan 

Tan 

Co  fan 

Tan 

Cotan 

o 

9.32747 

10.67253 

9.36336 

10.63664 

9.39677 

10.60323 

60 

1 

32810 

67190 

36394 

63606 

39731 

60269 

59 

2 

32872 

67128 

36452 

63548 

39785 

60?15 

68 

3 

32933 

67067 

36509 

63491 

39838 

60162 

57 

4 

32995 

67005 

36566 

63434 

89892 

60108 

66 

5 

33057 

66943 

36624 

68376 

89945 

60055 

65 

6 

33119 

66881 

36681 

63319 

39999 

60001 

54 

7 

83180 

66820 

36738 

63262 

40052 

59948 

53 

8 

83->42 

66758 

36795 

63205 

40106 

59894 

62 

9 

83303 

66697 

36852 

63148 

40159 

59841 

61 

10 

9.33365 

10.66635 

9.36909 

10.63091 

9.40212 

10.59788 

50 

n 

33426 

66574 

36966 

63034 

40266 

59734 

49 

12 

33487 

66513 

37023 

62977 

40319 

59681 

48 

13 

33548 

66452 

37080 

62920 

40372 

59628 

47 

14 

33609 

66391 

37137 

628C3 

40425 

59575 

46 

15 

336^0 

66330 

37193 

62807 

40478 

59522 

45 

16 

33731 

66269 

37250 

62750 

40531 

59469 

44 

17 

33792 

66208 

37306 

62694 

40584 

59416 

43 

18 

33853 

66147 

37363 

62637 

40636 

59364 

42 

19 

83913 

66087 

37419 

62581 

40689 

59311 

41 

20 

9.33974 

10.66026 

9.37476 

10.62524 

9.40742 

10.59258 

40 

21 

34034 

65966 

37532 

62468 

40795 

69205 

39 

22 

34095 

65905 

37588 

62412 

40847 

59153 

38 

23 

84155 

65845 

37644 

62356 

40900 

59100 

37 

24 

34215 

65785 

37700 

62300 

40952 

59048 

36 

25 

34276 

66724 

377'56 

62244 

41005 

58995 

35 

26 

34336 

65664 

37812 

62188 

41057 

58943 

34 

27 

34396 

65604 

37868 

62132 

41109 

58891 

33 

28 

|I456 

65544 

87924 

62076 

41161 

58839 

3d 

29 

34516 

65484 

87980 

62020 

41214 

58786 

31 

30 

9.34576 

10.65424 

9.38035 

10.61965 

9.41266 

10.58734 

30 

31 

34635 

65365 

38091 

61909 

41318 

58682 

29 

32 

34695 

65305 

38147 

61853 

41370 

58630 

28 

33 

34755 

65245 

38202 

61798 

41422 

58578 

27 

34 

34814 

65186 

38257 

61743 

41474 

58526 

26 

35 

34874 

65126 

38313 

61687 

41526 

68474 

25 

86 

34933 

65067 

38368 

61632 

41578 

58422 

24 

37 

34992 

65008 

38423 

61577 

41629 

58371 

23 

38 

35051 

64949 

38479 

61521 

41681 

58319 

22 

39 

35111 

64889 

38534 

61466 

41733 

58267 

21 

40 

9.35170 

10.64830 

9.38589 

10.61411 

9.41784 

10.58216 

20 

41 

35229 

64771 

38644 

61356 

41836 

58164 

19 

42 

35288 

64712 

38699 

61301 

41887 

58113 

18 

43 

35347 

64653 

38754 

61246 

41939 

58061 

17 

44 

35405 

64595 

38808 

61192 

41990 

58010 

16 

45 

35464 

64536 

38863 

61137 

42041 

57959 

15 

46 

35523 

64477 

38918 

61082 

42093 

57907 

14 

47 

35581 

64419 

38972 

61028 

42144 

57856 

13 

48 

35640 

64360 

39027 

60973 

42195 

57805 

12 

49 

35698 

64302 

39082 

60918 

42246 

57754 

11 

50 

9.35757 

10.64243 

9.39136 

10.60864 

9.42297 

10.57703 

10 

51 

35815 

64185 

39190 

60810 

42348 

57652 

9 

52 

35873 

64127 

39245 

60755 

42399 

57601 

8 

53 

35931 

64069 

39299 

60701 

42450 

57550 

7 

54 

35989 

64011 

39353 

60647 

42501 

57499 

6 

55 

36047 

63953 

39407 

60593 

42552 

57448 

5 

56 

36105 

63895 

39461 

60539 

42603 

57397 

4 

57 

36163 

63837 

39515 

60485 

42653 

57347 

8 

58 

36221 

63779 

39569 

60431 

42704 

57296 

2 

59 

36279 

63721 

39623 

60377 

42755 

57245 

1 

60 

36336 

63664 

39677 

60323 

42805 

57195 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

i 

77° 

76° 

76*  - 

196       TABLE  9.— LOG.  TANGENTS  AND   COTANGENTS. 


0 

15° 

16° 

17° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.42805 

10.57195 

9.45750 

10.54250 

9.48534 

10.51466 

60 

1 

42856 

57144 

45797 

54203 

48579 

51421 

59 

2 

42906 

57094 

45845 

54155 

48624 

51376 

58 

3 

42957 

57043 

45892 

54108 

48669 

51331 

57 

4 

43007 

56993 

45940 

54060 

48714 

51286 

56 

5 

43057 

56943 

45987 

54013 

48759 

51241 

55 

6 

43108 

56892 

46035 

53965 

48804 

51196 

54 

7 

43158 

56842 

46082 

53918 

48849 

51151 

53 

8 

43208 

56792 

46130 

53870 

48894 

51106 

52 

9 

43258 

56742 

46177 

53823 

48939 

51061 

51 

10 

9.43308 

10.56692 

9.46224 

10.53776 

9.48984 

10.51016 

50 

11 

43358 

56642 

46271 

53729 

49029 

50971 

49 

12 

43408 

56592 

46319 

53681 

49073 

50927 

48 

18 

43458 

56542 

46366 

53634 

49118 

50882 

47 

14 

43508 

56492 

46413 

53587 

49163 

50837 

46 

15 

43558 

56442 

46460 

53540 

49207 

50793 

45 

16 

43607 

56393 

46507 

53493 

49252 

50748 

44 

17 

43657 

56343 

46554 

53446 

49296 

50704 

43 

18 

43707 

56293 

46601 

53399 

49341 

50659 

42 

19 

43756 

56244 

46648 

53352 

49385 

50615 

41 

20 

9.43806 

10.56194 

9.46694 

10.53306 

9.49430 

10.50570 

40 

21 

43855 

56145 

46741 

53259 

49474 

50526 

39 

22 

43905 

56095 

46788 

53212 

49519 

50481 

38 

23 

43954 

56046 

46835 

53165 

49563 

50437 

37 

24 

44004 

55996 

46881 

53119 

49607 

50393 

36 

25 

44053 

55947 

46928 

53072 

49652 

50348 

85 

26 

44102 

55898 

46975 

53025 

49696 

50304 

34 

27 

44151 

55849 

47021 

52979 

49740 

50260 

33 

28 

44201 

55799 

47068 

52932 

49784 

50216 

32 

29 

44250 

55750 

47114 

52886 

49828 

50172 

31 

30 

9.44299 

10.55701 

9.47160 

10.52840 

9.49872 

10.50128 

30 

31 

44348 

55652 

47207 

52793 

49916 

50084 

29 

32 

44397 

55603 

47253 

52747 

49960 

50040 

28 

33 

44446 

55554 

47299 

52701 

50004 

49996 

27 

34 

44495 

55505 

47346 

52654 

50048 

49952 

26 

35 

44544 

55456 

47392 

52608 

50092 

49908 

25 

36 

44592 

55408 

47438 

52562 

50136 

49864 

24 

37 

44641 

55359 

47484 

52516 

50180 

49820 

23 

38 

44690 

55310 

47530 

52470 

50223 

49777 

22 

39 

44738 

55262 

47576 

52424 

50267 

49733 

21 

40 

9.44787 

10.55213 

9.47622 

10.52378 

9.50311 

10.49689 

20 

41 

44836 

55164 

47668 

52332 

50355 

49645 

19 

42 

44884 

55116 

47714 

52286 

50398 

49602 

18 

43 

44933 

55067 

47760 

52240 

50442 

49558 

17 

44 

44981 

55019 

47806 

52194 

50485 

49515 

16 

45 

45029 

54971 

47852 

52148 

50529 

49471 

15 

46 

45078 

54922 

47897 

52103 

50572 

49428 

14 

47 

45126 

54874 

47943 

52057 

50616 

49384 

13 

48 

45174 

54826 

47989 

52011 

50659 

49341 

12 

49 

45222 

54778 

48035 

51965 

50703 

49297 

11 

50 

9.45271 

10.54729 

9.48080 

10.51920 

9.50746 

10.49254 

10 

51 

45319 

54681 

48126 

51874 

50789 

49211 

9 

52 

45367 

54633 

48171 

51829 

50833 

49167 

8 

53 

45415 

54585 

48217 

51783 

50876 

49124 

7 

54 

45463 

54537 

48262 

51738 

50919 

49081 

6 

55 

45511 

54489 

48307 

51693 

50962 

49038 

5 

56 

45559 

54441 

48353 

51647 

51005 

48995 

4 

57 

45606 

54394 

48398 

51602 

51048 

48952 

3 

58 

45654 

54346 

48443 

51557 

51092 

48908 

2 

59 

457C2 

54298 

48489 

51511 

51135 

48865 

1 

60 

45750 

54250 

48534 

51466 

51178 

48822 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

74° 

73° 

72° 

TABLE  9.— LOG.  TANGENTS   AND  COTANGENTS.  197 


18° 

19° 

20° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.51178 

10.48822 

9.53697 

10.46303 

9.56107 

10.43893 

60 

1 

51221 

48779 

53738 

46262 

56146 

43854 

59 

2 

51264 

48736 

53779 

46221 

56185 

43815 

58 

3 

51306 

48694 

53820 

46180 

56224 

43776 

57 

4 

51349 

48651 

53861 

46139 

56264 

43736 

56 

5 

51392 

48608 

53902 

46098 

56303 

43697 

55 

6 

51435 

48565 

53943 

46057 

56342 

43658 

54 

7 

51478 

48522 

53984 

46016 

56381 

43619 

53 

8 

51520 

48480 

54025 

45975 

56420 

48580 

52 

9 

51563 

48437 

54065 

45935 

56459 

43541 

51 

10 

9.51606 

10.48394 

9.54106 

10.45894 

9.56498 

10.43502 

50 

11 

51648 

48352 

54147 

45853 

"  56537 

43463 

49 

12 

51691 

48309 

54187 

45813 

56576 

43424 

48 

13 

51734 

48266 

54228 

45772 

56815 

43385 

47 

14 

51776 

48224 

54269 

45731 

56654 

445346 

46 

15 

51819 

48181 

54309 

45691 

56693 

43307 

45 

16 

51861 

48139 

54350 

45650 

56738 

43268 

44 

17 

51903 

48097 

54390 

45610 

56771 

43229 

43 

18 

51946 

48054 

54431 

45569 

56810 

43190 

42 

19 

51988 

48012 

54471 

45529 

56849 

43151 

41 

20 

9.52031 

10.47969 

9.54512 

10.45488 

9.56887 

10.43113 

40 

21 

52073 

47927 

54552 

45448 

56926 

43074 

39 

22 

52115 

47885 

54593 

45407 

56965 

43035 

38 

23 

52157 

47843 

54633 

45367 

57004 

42996 

37 

24 

52200 

47800 

54673 

45327 

57042 

42958 

36 

25 

52242 

47758 

54714 

45286 

57081 

42919 

35 

26 

52284 

47716 

54754 

45246 

57120 

42880 

34 

27 

52326 

47674 

54794 

45206 

57158 

42842 

33 

28 

52368 

47632 

54835 

45165 

57197 

42803 

82 

29 

5241JJ 

47590 

54875 

45125 

57235 

42765 

31 

30 

9.52452 

10.47548 

9.54915 

10.45085 

9.57274 

10.42726 

30 

31 

52494 

47506 

54955 

45045 

57312 

42688 

29 

32 

52536 

47464 

54995 

45005 

57351 

42649 

88 

33 

52578 

47422 

55035 

44965 

57389 

42611 

27 

34 

52620 

47380 

55075 

44925 

57428 

42572 

26 

35 

52661 

47339 

55115 

44885 

57466 

42534 

25 

36 

52703 

47297 

55155 

44845 

57504 

42496 

24 

37 

52745 

47255 

55195 

44805 

57543 

42457 

23 

38 

52787 

47213 

55235 

44765 

57581 

42419 

22 

39 

52829 

47171 

55275 

44725 

57619 

42381 

21 

40 

9.52870 

10.47130 

9.55315 

10.44685 

9.57658 

10.42342 

20 

41 

52912 

47088 

55355 

44645 

57696 

42304 

19 

42 

52953 

47047 

55395 

44605 

57734 

42266 

18 

43 

52995 

47005 

55434 

44566 

57772 

42228 

17 

44 

53037 

46963 

55474 

44526 

57810 

42190 

16 

45 

53078 

46922 

55514 

44486 

57849 

42151 

15 

46 

53120 

46880 

55554 

44446 

57887 

42113 

14 

47 

53161 

46839 

55593 

44407 

57925 

42075 

13 

48 

53202 

46798 

55633 

44367 

57963 

42037 

12 

49 

53244 

46756 

55673 

44327 

56001 

41999 

11 

50 

9.53285 

10.46715 

9.55712 

10.44288 

9.58039 

10.41961 

10 

51 

53327 

46673 

55752 

44248 

58077 

41923 

9 

52 

53368 

46632 

55791 

44209 

58115 

41885 

8 

53 

53409 

46591 

55831 

44169 

58153 

41847 

7 

54 

53450 

46550 

55870 

44130 

58191 

41809 

6 

55 

53492 

46508 

55910 

44090 

58229 

41771 

5 

56 

53533 

46467 

55949 

44051 

58267 

41733 

4 

57 

53574 

46426 

55989 

44011 

58304 

41696 

3 

58 

53615 

46385 

56028 

43972 

58342 

41658 

2 

59 

53656 

•  46344 

56067 

43933 

58380 

41620 

1 

60 

53697 

46303 

56107 

43893 

58418 

41582 

0 

> 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

71° 

70° 

69° 

198       TABLE  9.— LOG.  TANGENTS  AND  COTANGENTS, 


1 

21° 

22° 

23° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.58418 

10.41582 

9.60641 

10.39359 

9.62785 

10.37215 

60 

1 

58455 

41545 

60677 

39323 

62820 

37180 

59 

2 

58493 

41507 

60714 

39-386 

62855 

37145 

58 

3 

58531 

41469 

60750 

39250 

62890 

37110 

57 

4 

58569 

41431 

60786 

39214 

62926 

37074 

56 

5 

58606 

41394 

60823 

39177 

62961 

37039 

55 

6 

58644 

41356 

60859 

39141 

62996 

37004 

54 

7 

58681 

41319 

60895 

39105 

63031 

36.969 

53 

8 

58719 

41281 

60931 

39069 

68066 

36934 

52 

9 

58757 

41243 

60967 

39033 

63101 

36899 

51 

10 

9.58794 

10.41206 

9.61004 

10.38996 

9.63135 

10.36865 

50 

11 

58832 

41168 

61040 

38960 

63170 

36830 

49 

12 

58869 

41131 

61076 

38924 

63205 

36795 

•  48 

13 

58907 

41093 

61112 

38888 

63240 

36760 

47 

14 

58944 

41056 

61148 

38852 

63275 

36725 

46 

15 

58981 

41019 

61184 

38816 

63310 

36690 

45 

16 

59019 

40981 

61220 

38780 

68345 

36655 

44 

17 

59056 

40944 

61256 

38744 

63379 

36621 

43 

18 

59094 

40906 

61292 

38708 

63414 

36586 

42 

19 

59131 

40869 

61328 

38672 

63449 

36551 

41 

20 

9.59168 

10.40882 

9.61304 

10.38636 

9.63484 

10.36516 

40 

21 

69205 

40795 

61400 

3860C 

63519 

36481 

39 

22 

59243 

40757 

61436 

3856* 

63553 

36447 

88 

23 

59280 

40720 

61472 

38528 

63588 

36412 

37 

24 

59317 

40688 

61508 

38492 

63623 

36377 

36 

25 

59854 

40646 

61544 

38456 

63657 

36343 

85 

26 

59391 

40609 

61579 

38421 

63692 

36308 

34 

27 

59429 

40571 

61615 

38385 

G3726 

36274 

33 

28 

59466 

40534 

61651 

38349 

63761 

36239 

32 

29 

59503 

40497 

61687 

38313 

63796 

36204 

31 

80 

9.59540 

10.40460 

9.61722 

10.38278 

9.63830 

10.36170 

30 

31 

59577 

40423 

61758 

38242 

63865 

36135 

29 

32 

59614 

40386 

61794 

38206 

63899 

36101 

28 

33 

59651 

40349 

61830 

38170 

63934 

36066 

27 

34 

69688 

40312 

61865 

38135 

63968 

36032 

26 

35 

59725 

40275 

61901 

38099 

64003 

35997 

25 

36 

69762 

40238 

61936 

38064 

64037 

35963 

24 

87 

59799 

40201 

61972 

38028 

64072 

35928 

23 

38 

59835 

40165 

62008 

37992 

64106 

35894 

22 

39 

59872 

40128 

62043 

37957 

64140 

35860 

21 

40 

9.59909 

10.40091 

9.62079 

10.37921 

9.64175 

10.35825 

20 

41 

59946 

40054 

62114 

87886 

64209 

35791 

19 

42 

59983 

40017 

62150 

37850 

64243 

35757 

18 

43 

60019 

89981 

62185 

37815 

64278 

35722 

17 

44 

60056 

39944 

62221 

37779 

64312 

35688 

16 

45 

60093 

39907 

62256 

37744 

64346 

35654 

15 

46 

60130 

39870 

62292 

37708 

64381 

35619 

14 

47 

60166 

39834 

62327 

37673 

64415 

35585 

13 

48 

60203 

39797 

62362 

37638 

64449 

35551 

12 

49 

60240 

39760 

62398 

37602 

64483 

35517 

11 

50 

9.60276 

10.39724 

9.62433 

10.37567 

9.64517 

10.85483 

10 

51 

60313 

39687 

62468 

37532 

64552 

35448 

9 

52 

60349 

39651 

62504 

37496 

64586 

35414 

8 

53 

60386 

39614 

62539 

37461 

64620 

35380 

IJr 

54 

60422 

39578 

62574 

37426 

64654 

35346 

6 

55 

60459 

39541 

62609 

37391 

64688 

35312 

5 

56 

60495 

39505 

62645 

37855 

64722 

35278 

4 

57 

60532 

39468 

62680 

37320 

64756 

35244 

8 

58 

605(58 

39432 

62715 

37285 

64790 

35210 

59 

60605 

89395 

62750 

37250 

64824 

35176 

1  1 

60 

60641 

39359 

62785 

87215 

64858 

85142 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

f 

68» 

67° 

66° 

TA!BLE  9.— LOG.  TANGENTS   AND   COTANGENTS.  199 


/ 

24° 

25° 

26° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.64858 

10.35142 

9.66867 

10.33133 

9.68818 

10.31182 

60 

1 

64892 

35108 

66900 

33100 

68850 

31150 

59 

2 

64926 

35074 

66933 

33067 

68882 

31118 

58 

3 

64960 

35040 

66966 

33034 

68914 

31086 

57 

4 

64994 

35006 

66999 

33001 

68946 

31054 

56 

5 

65028 

84972 

67032 

32968 

68978 

31022 

55 

6 

65062 

34938 

67065 

32935 

69010 

90990 

54 

7 

65096 

34904 

67098 

32902 

69042 

30958 

53 

8 

65130 

34870 

67131 

32869 

69074 

30926 

52 

9 

65164 

34836 

67163 

32837 

69106 

30894 

51 

10 

9.65197 

10.34803 

9.67196 

10.32804 

9-.  69138 

10.30862 

50 

11 

65231 

34769 

67229 

32771 

69170 

30830  i  49 

12 

65265 

34735 

67262 

32738 

69202 

80798 

48 

13 

65299 

34701 

67295 

32705 

69234 

30766 

47 

14 

65333 

34667 

67327 

32673 

69266 

80734 

46 

15 

65366 

34634 

67360 

32640 

69298 

30702 

45 

16 

65400 

34600 

67393 

32607 

69329 

30671 

44 

17 

65434 

34566 

67426 

82574 

69361 

30689 

43 

18 

65467 

34533 

67458 

32542 

69393 

30607 

42 

19 

65501 

34499 

67491 

32509 

69425 

30575 

41 

20 

9.65535 

10.34465 

9.67524 

10.32476 

9.69457 

10.30543 

40 

21 

65568 

34432 

67556 

32444 

69488 

30512 

39 

22 

65602 

34398 

67589 

32411 

69520 

30480 

38 

23 

65636 

34364 

67622 

32378 

69552 

30448 

37 

24 

656(59 

34331 

67654 

32346 

69584 

30416 

36 

25 

65703 

34297 

67687 

32313 

69615 

30385 

35 

26 

65736 

34264 

67719 

32281 

69647 

30353 

34 

27 

65770 

34230 

67752 

32248 

69679 

30321 

33 

28 

65803 

^  34197 

67785 

32215 

69710 

30290 

32 

29 

65837 

34163 

67817 

32183 

69742 

30258 

31 

30 

9.65870 

10.34130 

9.67850 

10.32150 

9.69774 

10.30226 

30 

31 

65904 

34096 

67882 

32118 

69805 

30195 

29 

32 

65937 

34063 

67915 

32085 

69837 

30163 

28 

33 

65971 

34029 

67947 

32053 

69868 

30132 

27 

34 

66004 

33996 

67980 

32020 

69900 

30100 

26 

35 

66038 

33962 

68012 

31988 

69932 

80068 

25 

M 

66071 

83929 

68044 

31956 

69963 

30037 

24 

37 

66104 

33896 

68077 

31923 

69995 

30005 

23 

38 

66138 

33862 

68109 

31891 

70026 

29974 

22 

39 

66171 

33829 

68142 

31858 

70058 

29942 

21 

40 

9.66204 

10.38796 

9.68174 

10.31826 

9.70089 

10.29911 

20 

41 

66238 

33762 

68206 

31794 

70121 

29879 

19 

42 

66271 

33729 

68239 

31761 

70152 

29848 

18 

43 

66304 

33696 

68271 

31729 

70184 

29816 

17 

44 

66337 

33663 

68303 

31697 

70215 

29785 

16 

45 

66371 

33629 

68336 

31664 

70247 

29753 

15 

46 

66404 

33506 

68368 

31632 

70278 

29722 

14 

47 

66437 

33563 

68400 

31600 

70309 

29691 

13 

48 

66470 

33530 

68432 

31568 

70341 

29659 

12 

49 

06503 

33497 

68465 

31585 

70372 

29628 

11 

50 

0.0(5537 

10.33463 

9.68497 

10.31503 

9.70404 

10.29596 

10 

51 

66570 

33430 

6a529 

31471 

70435 

29565 

9 

52 

66603 

33397 

68561 

31439 

70466 

29534 

8 

f>3 

66636 

33364 

68593 

31407 

70498 

29502 

7 

54 

66669 

33331 

68626 

31374 

70529 

29471 

6 

55 

66702 

33298 

68658 

31342 

70560 

29440 

5 

56 

6673") 

33265 

68690 

31310 

70592 

29408 

4 

57 

66768 

33232 

68722 

31278 

70623 

29377 

8 

58 

66801 

33199 

68754 

31246 

70654 

29346    2 

59 

66834 

33166 

68786 

31214 

70685 

29315  !   1 

fiO     66867 

33133 

68818 

31182 

70717 

29283  i   0 

/ 

Otan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

65° 

64° 

63° 

200       TABLE  9.— LOG.  TANGENTS  AND  COTANGENTS. 


/ 

27° 

28° 

29° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.70717 

10.29283 

9.72567 

10.27433 

9.74375 

10.25625 

60 

1 

70748 

29252 

72598 

27402 

74405 

25595 

59 

2 

70779 

29221 

72628 

27372 

74435 

25565 

58 

3 

70810 

29190 

72659 

27341 

74465 

25535 

57 

4 

70841 

29159 

72689 

27311 

74494 

25506 

56 

5 

70873 

29127 

72720 

27280 

74524 

25476 

55 

6 

70904 

29096 

72750 

27250 

74554 

25446 

54 

7 

70935 

29065 

72780 

27220 

74583 

25417 

53 

8 

70966 

29034 

72811 

27189 

74613 

25387 

52 

9 

70997 

29003 

72841 

27159 

74643 

25357 

51 

10 

9.71028 

10.28978 

9.72872 

10.27128 

9.74673 

10.25327 

50 

11 

71059 

28941 

72902 

27098 

74702 

25298 

49 

12 

71090 

28910 

72932 

27068 

74732 

25268 

48 

13 

71121 

28879 

72963 

27037 

74762 

25238 

47 

14 

71153 

28847 

72993 

27007 

74791 

25209 

46 

15 

71184 

28816 

73023 

26977 

74821 

25179 

45 

16 

71215 

28785 

73054 

26946 

74851 

25149 

44 

17 

71246 

28754 

73084 

26916 

74880 

25120 

43 

18 

71277 

28723 

73114 

26886 

74910 

25090 

42 

19 

71308 

28692 

73144 

26856 

74939 

25061 

41 

20 

9.71339 

10.28661 

9.73175 

10.26825 

9.74969 

10.25031 

40 

21 

71370 

28630 

73205 

26795 

74998 

25002 

39 

22 

71401 

28599 

73235 

26765 

75028 

24972 

38 

23 

71431 

28569 

73265 

26735 

75058 

24942 

37 

24 

71462 

28538 

73295 

26705 

75087 

24913 

36 

25 

71493 

28507 

73326 

26674 

75117 

24883 

35 

26 

71524 

28476 

73356 

26644 

75146 

24854 

34 

27 

71555 

28445 

73386 

26614 

75176 

24824 

33 

28 

71586 

28414 

73416 

26584 

75205 

24795 

32 

29 

71617 

28383 

73446 

26554 

75235 

24765 

31 

30 

9.71648 

10.28352 

9.73476 

10.26524 

9.75264 

10.24736 

30 

31 

71679 

28321 

73507 

26493 

75294 

24706 

29 

32 

71709 

28291 

73537 

26463 

75323 

24677 

28 

33 

71740 

28260 

73567 

26433 

75353 

24647 

27 

34 

71771 

28229 

73597 

26403 

75382 

24618 

26 

35 

-  71802 

28198 

73627 

26373 

75411 

24589 

25 

36 

71833 

28167 

73657 

26343 

75441 

24559 

24 

87 

71863 

28137 

73687 

26313 

75470 

24530 

23 

38 

71894 

28108 

73717 

26283 

75500 

24500 

22 

39 

71925 

28075 

73747 

26253 

75529 

24471 

21 

40 

9.71955 

10.28045 

9.73777 

10.26223 

9.75558 

10.24442 

20 

41 

71986 

28014 

73807 

26193 

75588 

24412 

19 

42 

72017 

27983 

73837 

26163 

75617 

24383 

18 

43 

72048 

27952 

73867 

26133 

75647 

24353 

17 

44 

72078 

27922 

73897 

26103 

75676 

24324 

16 

45 

72109 

27891 

78927 

26073 

75705 

24295 

15 

46 

72140 

27860 

73957 

26043 

75735 

24265 

14 

47 

72170 

27830 

73987 

26013 

75764 

24236 

13 

48 

72201 

27799 

74017 

25983 

75793 

24207 

12 

49 

72231 

27769 

74047 

25953 

75822 

24178 

11 

50 

9.72262 

10.27738 

9.74077 

10.25923 

9.75852 

10.24148 

10 

51 

72293 

27707 

74107 

25893 

75881 

24119 

9 

52 

72323 

27677 

74137 

25863 

75910 

24080 

8 

53 

72354 

27646 

74166 

25834 

75939 

24061 

7 

54 

72384 

27616 

74196 

25804 

75969 

24031 

6 

55 

72415 

27585 

74226 

25774 

75998 

24002 

5 

56 

72445 

27555 

74256 

25744 

76027 

23973 

4 

57 

72476 

27524 

74286 

25714 

76056 

23944 

3 

58 

72506 

27494 

74316 

25684 

76086 

28914 

2 

59 

72537 

27463 

74345 

25655 

76115 

23885 

1 

60 

72567 

27433 

74375 

25625 

76144 

23856 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

62  • 

, 

61' 

60° 

TABLE  9.— LOG.  TANGENTS  AND  COTANGENTS.  201 


f 

30° 

31° 

32° 

t 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.76144 

10.23856 

9.77877 

10.22123 

9.79579 

10.20421 

60 

1 

76173 

23827 

77906 

22094 

79607 

20393 

59 

2 

76202 

23798 

77935 

22065 

79635 

20365 

58 

3 

76231 

23769 

77963 

22037 

79663 

20337 

57 

4 

76261 

23739 

77992 

22008 

79691 

20309 

56 

5 

76290 

23710 

78020 

21980 

797J  9 

20281 

55 

6 

76319 

23681 

78049 

21951 

79747 

20253 

64 

7 

76348 

23C52 

78077 

21923 

79776 

20224 

53 

8 

76377 

23623 

78106 

21894 

79804 

20196 

52 

9 

76406 

23594 

78135 

.21865 

.  79832 

20168 

51 

10 

9.76435 

10.23565 

9.78163 

10.21837 

9.79860 

10.20140 

50 

11 

76464 

23536 

78192 

21808 

79888 

20112 

49 

12 

76493 

23507 

78220 

21780 

79916 

20084 

48 

13 

76522 

23478 

78249 

21751 

79944 

20056 

47 

14 

76551 

23449 

78277 

21723 

79972 

20028 

46 

15 

76580 

23420 

78306 

21694 

80000 

20000 

45 

16 

76609 

23391 

78334 

21666 

80028 

19972 

44 

17 

76639 

23361 

78363 

21637 

80056 

19944 

43 

18 

76668 

23332 

78391 

21609 

80084 

19916 

42 

13 

76697 

23303 

78419 

21581 

80112 

19888 

41 

20 

9.76725 

10.23275 

9.78448 

10.21552 

9.80140 

10.19860 

40 

21 

76754 

23246 

78476 

21524 

80168 

19832 

39 

22 

76783 

23217 

78505 

21495 

80195 

19805 

38 

23 

76812 

23188 

78533 

21467 

80223 

19777 

87 

24 

76841 

2M59 

78562 

21438 

80251 

19749 

86 

25 

76870 

23180 

78590 

21410 

80279 

19721 

35 

26 

76899 

23101 

78618 

21382 

80307 

19693 

34 

27 

76928 

23072 

78647 

21353 

80335 

19665 

33 

28 

76957 

23043 

78675 

21325 

80363 

19637 

82 

29 

76986 

23014 

78704 

21296 

80391 

19609 

31 

30 

9.77015 

10.22985 

9.78732 

10.21268 

9.80419 

10.19581 

30 

31 

77044 

22956 

78760 

21240 

80447 

19553 

29 

32 

77073 

22927 

78789 

21211 

80474 

19526 

28 

33 

77101 

22899 

78817 

21183 

80502 

19498 

27 

31 

77130 

22870 

78845 

21155 

80530 

19470 

26 

35 

77159 

22841 

78874 

21126 

80558 

19442 

25 

36 

77188 

22812 

78902 

21098 

80586 

19414 

24 

37 

77217 

22783 

78930 

21070 

80614 

19386 

23 

38 

77246 

22754 

78959 

21041 

80642 

19358 

22 

39 

77274 

22726 

78987 

21013 

80669 

19331 

21 

40 

9.77303 

10.22697 

9.79015 

10.20985 

9.80697 

10.19303 

20 

41 

77332 

22668 

79043 

20957 

80725 

19275 

19 

42 

77361 

22639 

79072 

20928 

80753 

19247 

18 

43 

77390 

22610 

79100 

20900 

80781 

19219 

17 

44 

77418 

22582 

79128 

20872 

80808 

19192 

16 

45 

77447 

22553 

79156 

20844 

80836 

19164 

15 

46 

77476 

22524 

79185 

20815 

80864 

19136 

14 

47 

77505 

22495 

79213 

20787 

80892 

19108 

13 

48 

77533 

22467 

79241 

20759 

80919 

19081 

12 

49 

77562 

22438 

79269 

20731 

80947 

19053 

11 

50 

9.77591 

10.22409 

9.79297 

10.20703 

9.80975 

10.19025 

10 

51 

77619 

22381 

79326 

20674 

81003 

18997 

9 

52 

77648 

22352 

79354 

20646 

81030 

18970 

8 

53 

77677 

22323 

79382 

20618 

81058 

18942 

7 

54 

77706 

22294 

79410 

20590 

81086 

18914 

6 

55 

77734 

22266 

79438 

20562 

81113 

18887 

5 

50 

77763 

22237 

79466 

20534 

81141 

18859 

4 

57 

77791 

22209 

79495 

20505 

81169 

18831 

3 

58 

77820 

22180 

79523 

20477 

81196 

18804 

2 

59 

77849 

22151 

79551 

20449 

81224 

18776 

1 

60 

77877 

22123 

79579 

20421 

81252 

18748 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

59° 

68° 

67° 

202       TABLE  9.— LOG.  TANGENTS   AND   COTANGENTS. 


/ 

83° 

84° 

85° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.81252 

10.18748 

9.82899 

.  10.17101 

9.84523 

10.15477 

60 

1 

81279 

18721 

82926 

17074 

81550 

15450 

59 

2 

81307 

18693 

82953 

17047 

84576 

15424 

58 

3 

81335 

18665 

82980 

17020 

84603 

15397 

57 

4 

81362 

18638 

83008 

16992 

84630 

15870 

56 

5 

81390 

18610 

83035 

16965 

84657 

15343 

55 

6 

81418 

18582 

83063 

16938 

84684 

15316 

54 

7 

81445 

18555 

83089 

16911 

84711 

15289 

53 

8 

81473 

18527 

83117 

16883 

84738 

15262 

52 

9 

81500 

18500 

83144 

16856 

84764 

15236 

51 

10 

9.81528 

10.18472 

9.83171 

10.168-29 

9.84791 

10.15209 

50 

11 

81556 

18444 

83198 

16802 

84818 

15182 

49 

12 

81583 

18417 

83-225 

16775 

84845 

15155 

48 

18 

81611 

18389 

83252 

16748 

84872 

15128 

47 

14 

81638 

18362 

83280 

16720 

84899 

15101 

46 

15 

81666 

18334 

83307 

16693 

84925 

15075 

45 

16 

81693 

18307 

83334 

16666 

84952 

15048 

44 

17 

81721 

18279 

83361 

16639 

84979 

15021 

43 

18 

81748 

18252 

83388 

16612 

85006 

14994 

43 

19 

81776 

18224 

83415 

16585 

85033 

14967 

41 

20 

9.81803 

10.18197 

9.83142 

10.16558 

9.85059 

10.14941 

40 

21 

81831 

18169 

83470 

16530 

85086 

14914 

39 

22 

81858 

18142 

83497 

16503 

85113 

14887 

38 

23 

81886 

18114 

83524 

16476 

85140 

14860 

37 

24 

81913 

18087 

83551 

16449 

85166 

14834 

36 

25 

81941 

18059 

83578 

16423 

85193 

14807 

35 

26 

81968 

18032 

83605 

16395 

85220 

14780 

34 

27 

81996 

18004 

83632 

16368 

85247 

14753 

33 

28 

82023 

17'977 

83659 

16341 

85273 

14737 

3.' 

29 

82051 

17949 

83686 

16314 

85300 

14700 

81 

30 

9.82078 

10.17922 

9.83713 

10.16287 

9.85327 

10.14673 

30 

31 

82106 

17894 

83740 

16260 

85354 

14046 

29 

32 

82133 

178«7 

83768 

16232 

85380 

14620 

*8 

83 

82161 

17839 

83795 

16205 

85407 

14593 

34 

82188 

17812 

83822 

16178 

85434 

145G6 

2(j 

35 

82215 

17785 

83849 

16151 

85460 

14540 

25 

36 

82243 

17757 

83876 

16124 

85487 

14513 

24 

37 

82270 

17730 

83903 

16097 

85514 

14486 

23 

38 

82298 

17702 

83930 

16070 

85540 

14460 

22 

39 

82325 

17675 

83957 

16043 

£5567 

14433 

21 

40 

9.82352 

10.17648 

9.83984 

10.16016 

9.85594 

10.14406 

20 

41 

82380 

17620 

84011 

15989 

85620 

14380 

19 

42 

82407 

17593 

84038 

15962 

85647 

14353 

18 

43 

82435 

17565 

84065 

15935 

85674 

14326 

17 

44 

82462 

17538 

8409-2 

15908 

85700 

14300 

16 

45 

82489 

17511 

84119 

15881 

85727 

14273 

15 

46 

82517 

17483 

84146 

15854 

85754 

14246 

14 

47 

82544 

17456 

84173 

15827 

85780 

14220 

18 

48 

82571 

17429 

84200 

15800 

85807 

14193 

13 

49 

82599 

17401 

84227 

15773 

85834 

141C6 

11 

50 

9.82626 

10.17374 

9.84254 

10.15746 

9.85860 

10.14140 

10 

51 

82653 

17347 

84280 

15720 

85887 

14113 

9 

52 

82681 

17319 

84307 

15693 

85913 

14087 

8 

53 

82708 

17292 

84334 

15666 

85940 

14060 

7 

54 

82735 

.  17265 

84361 

15639 

85967 

14033 

6 

55 

82762 

17238 

84388 

15612 

85993 

14007 

5 

56 

82790 

17210 

84415 

15585 

86020 

13980 

4 

57 

82817 

17183 

84443 

15558 

86046 

13954 

3 

58 

83844 

17156 

84469 

15531 

86073 

13927 

2 

59 

82871 

17129 

84496 

15504 

86100 

13900 

1 

60 

82899 

17101 

84523 

15477 

86126 

13874 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

56° 

65° 

64° 

TABLE  9.— LOG.  TANGENTS  AND  COTANGENTS.  203 


/ 

80° 

87° 

88° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.86126 

10.13874 

9.87711 

10.12289 

9.89281 

10.10-719 

60 

1 

86153 

13847 

87738 

12262 

89307 

10693 

59 

2 

86179 

13821 

87764 

12236 

89333 

10667 

58 

3 

86206 

13794 

87790 

12210 

89359 

10641 

57 

4 

86232 

13768 

87817 

12183 

89385 

10615 

56 

5 

86259 

13741 

87843 

12157 

89411 

10589 

55 

6 

86285 

13715 

87869 

12131 

89437 

10563 

54 

7 

86312 

13688 

87895 

12105 

89463 

10537 

53 

8 

8633S 

13662 

87922 

12078 

89489 

10511 

52 

9 

86365 

13635 

87948 

12052 

•  89515 

10485 

61 

10 

9.86392 

10.13608 

9.87974 

10.12026 

9.89541 

10.10459 

50 

11 

86418 

13582 

88000 

12000 

89567 

10433  I  49 

12 

86445 

13555 

88027 

11973 

89593 

10407    48 

13 

86471 

13529 

88053 

11947 

89619 

10381    47 

14 

86498 

13502 

88079 

11921 

89645 

10355    46 

15 

86524 

13476 

88105 

11895 

89671 

10329 

45 

16 

86551 

13449 

88J31 

11869 

89697 

10303 

44 

17 

86577 

13423 

88158 

11842 

89723 

10277 

43 

18 

86603 

13397 

88184 

11816 

89749 

10251 

42 

19 

86630 

13370 

88210 

11790 

89775 

10225 

41 

20 

9.86656 

10.13344 

9.88236 

10.11764 

9.89801 

10.10199 

40 

21 

.  86683 

13317 

88262 

11738 

89827 

10173 

39 

22 

86709 

13291 

88289 

11711 

89853 

10147 

38 

23 

86736 

133£4 

88315 

11685 

89879 

10121 

37 

24 

86762 

13238 

88341 

11659 

89905 

10095 

36 

25 

86789 

13211 

88367 

11633 

89931 

10069 

35 

26 

86815 

13185 

88393 

11607 

89957 

10043 

34 

27 

86842 

13158 

88420 

11580 

89983 

10017 

33 

28 

86868 

13132 

88446 

11554 

90009 

09991 

32 

29 

86894 

13106 

88472 

11528 

90035 

09965 

31 

30 

9.86921 

10.13079 

9.88498 

10.11502 

9.90061 

10.09939 

30 

31 

86947 

13053 

88524 

11476 

90086 

09914 

29 

32 

86974 

13026 

88550 

11450 

90112 

09888 

28 

33 

87000 

13000 

88577 

11423 

90138 

09862 

27 

34 

87027 

12973 

88603 

11397 

90164 

09836 

26 

35 

87053 

12947 

88629 

11371 

90190 

09810 

25 

36 

87079 

12921 

88655 

11345 

90216 

09784 

24 

37 

87106 

12894 

88681 

11319 

90242 

09758 

23 

38 

87132 

12868 

88707 

11293 

90268 

09732 

22 

39 

87158 

12842 

88733 

11267 

90294 

09706 

21 

40 

9.87185 

10.12815 

9.88759 

10.11241 

9.90320 

10.09680 

20 

41 

87211 

12789 

88786 

11214 

90346 

09654 

19 

42 

87238 

12762 

88812 

11188 

90371 

09629 

18 

43 

87264 

12736 

88838 

11162 

90397 

09603 

17 

44 

87290 

12710 

88864 

11136 

90423 

09577 

16 

45 

87317 

12683 

88890 

11110 

90449 

09551 

15 

46 

87343 

12657 

88916 

11084 

90475 

09525 

14 

47 

87369 

12631 

88942 

11058 

90501 

09499 

13 

48 

87396 

12604 

88968 

11032 

90527 

09473 

12 

49 

87422 

12578 

88994 

11006 

90553 

09447 

11 

50 

9.87448 

10.12552 

9.89020 

10.10980 

9.90578 

10.09422 

10 

51 

87475 

12525 

89046 

10954 

90604 

09396 

9 

52 

87501 

12499 

89073 

10927 

90630 

09370 

8 

53 

87527 

12473 

89099 

10901 

90656 

09344 

7 

54 

87554 

12446 

89125 

10875 

90682 

09818 

6 

55 

87580 

12420 

89151 

10849 

90708 

09292 

5 

56 

87606 

12394 

89177 

10823 

90734 

09266 

4 

57 

87633 

12367 

89203 

10797 

90759 

09241 

3 

58 

87659 

12341 

89229 

10771 

90785 

09215 

2 

59 

87685 

12315 

89255 

10745 

90811 

09189 

1 

60 

87711 

12289 

89281 

10719 

90837 

09163 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

63° 

62° 

61' 

204       TABLE  9.— LOG.  TANGENTS  AND  COTANGENTS- 


89° 

40° 

41° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.90837 

10.09163 

9.92381 

10.07619 

9.93916 

10.06084 

60 

1 

90863 

09137 

92407 

07593 

93942 

06058 

59 

2 

90889 

09111 

92433 

07567 

93967 

06033 

58 

3 

90914 

09086 

92458 

07542 

93993 

06007 

57 

4 

90940 

09060 

92484 

07516 

94018 

05982 

56 

5 

90966 

09034 

92510 

07490 

94044 

05956 

55 

6 

90992 

09008 

92535 

07465 

94069 

05931 

54 

7 

91018 

08982 

92561 

07439 

94095 

05905 

53 

8 

91043 

08957 

92587 

07413 

94120 

05880 

52 

9 

91069 

08931 

92612 

07388 

94146 

05854 

51 

10 

9.91095 

10.08905 

9.92638 

10.07362 

9.94171 

10.05829 

50 

11 

91121 

08879 

92663 

07337 

94197 

05803 

49 

12 

91147 

08853 

92689 

07311 

94222 

05778 

48 

13 

91172 

08828 

92715 

07285 

94248 

05752 

47 

14 

91198 

08802 

92740 

07260 

94273 

05727 

46 

15 

9J224 

08776 

92766 

07234 

94299 

05701 

45 

16 

91250 

08750 

92792 

07208 

94324 

05676 

44 

17 

91276 

08724 

92817 

07183 

94350 

05650 

43 

18 

91301 

08699 

92843 

07157 

94375 

05625 

42 

19 

91327 

08673 

92868 

07132 

94401 

05599 

41 

20 

9.91353 

10.08647 

9.92894 

10.07106 

9.94426 

10.05574 

40 

21 

91379 

08621 

92920 

07080 

94452 

05548 

39 

22 

91404 

08596 

92945 

07055 

94477 

05523 

38 

23 

91430 

08570 

92971 

07029 

94503 

05497 

37 

24 

91456 

08544 

92996 

07004 

94528 

05472 

36 

25 

9148,! 

08518 

93022 

06978 

94554 

05446 

35 

26 

91507 

08493 

93048 

06952 

94579 

05421 

34 

27 

91533 

08467 

93073 

06927 

94604 

05396 

33 

28 

91559 

08441 

93099 

06901 

94G30 

05370 

32 

29 

91585 

08415 

93124 

06876 

94655 

05345 

3! 

30 

9.91610 

10.08390 

9.93150 

10.06850 

9.94681 

10.05319 

30 

31 

91636 

08364 

93175 

06825 

94706 

05294 

29 

33 

91662 

08338 

93201 

06799 

94732 

05268 

28 

33 

91683 

08312 

93227 

06773 

94757 

05243 

27 

34 

91713 

08287 

93252 

06748 

94783 

05217 

26 

35 

91739 

08261 

93278 

06722 

94808 

05192 

25 

36 

91765 

08235 

93303 

06697 

94834 

05166 

24 

37 

91791 

08209 

93329 

06671 

94859 

05141 

23 

38 

91816 

08184 

93354 

06646 

94884 

05116 

22 

39 

91842 

08158 

93380 

06620 

94910 

05090 

21 

40 

9.91868 

10.08132 

9.93406 

10.06594 

9.94935 

10.05065 

20 

41 

91893 

08107 

93431 

06569 

94961 

05039 

19 

4-4 

91919 

08081 

93457 

06543 

94986 

05014 

18 

43 

91945 

08055 

93482 

06518 

95012 

04988 

17 

44 

91971 

08029 

93508 

06492 

95037 

04963 

16 

45 

91996 

08004 

93533 

06467 

95062 

04938 

15 

46 

92022 

07978 

93559 

06441 

95088 

04912 

14 

47 

92048 

07952 

93584 

06416 

95113 

04887 

13 

48 

92073 

07927 

93610 

06390 

95139 

04861 

12 

49 

92099 

07901 

93636 

06364 

95164 

04836 

11 

50 

9.92125 

10.07875 

9.93661 

10.06339 

9.95190 

10.04810 

10 

51 

92150 

07850 

93687 

06318 

95215 

04785 

9 

52 

92176 

07824 

93712 

06288 

95240 

04760 

8 

53 

92202 

07798 

93738 

06262 

95266 

04734 

7 

54 

92227 

07773 

93763 

06237 

95291 

04709 

6 

55 

92253 

07747 

93789 

06211 

95317 

04683 

5 

56 

92279 

07721 

93814 

06186 

95342 

04658 

4 

57 

92304 

07696 

93840 

06160 

95368 

04632 

3 

58 

92330 

07670 

93865 

06135 

95393 

04607 

0 

59 

92356 

07644 

93891 

06109 

95418 

04582 

1 

60 

92381 

07619 

93916 

06084 

95444 

04556 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

60* 

49° 

48° 

TABLE  9.— LOG.  TANGENTS   AND  COTANGENTS.  205 


42° 

43° 

440 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.95444 

10.04556 

9.96966 

10.03034 

9.98484 

10.01516 

60 

1 

95469 

04531 

96991 

03009 

98509 

01491 

59 

2 

95495 

04505 

97016 

02984 

98534 

01466 

58 

3 

95520 

04480 

97042 

0-^958 

98560 

01440 

57 

4 

95545 

04455 

97067 

02933 

98585 

01415 

56 

5 

95571 

04429 

97092 

02908 

98610 

01390 

55 

6 

95596 

04404 

97118 

02882 

98635 

01365 

54 

95622 

04378 

97143 

02857 

98661 

01339 

53 

8 

95647 

04353 

97168 

02832 

98686 

01314 

52 

9 

95672 

04328 

97193 

02807 

98711 

01289 

51 

10 

9.95698 

10.04302 

9.97219 

10.02781 

9.98737 

10.01263 

50 

11 

95723 

04277 

97244 

02756 

98762 

01238 

49 

12 

95748 

04252 

97269 

02731 

98787 

01213 

48 

13 

95774 

04226 

97295 

02705 

98812 

01188 

47 

14 

95799 

04201 

97320 

02680 

98838 

01162 

46 

15 

95825 

04175 

97345 

02655 

98863 

01137 

45 

16 

95850 

04150 

97371 

02629 

98888 

01112 

44 

1? 

95875 

04125 

97396 

02604 

98913 

01087 

43 

18 

95901 

04099 

97421 

02579 

98939 

01061 

42 

19 

95926 

04074 

97447 

02553 

98964 

01086 

41 

20 

9.95952 

10.04048 

9.97472 

10.02508 

9.98989 

10.01011 

40 

21 

95977 

04023 

97497 

02503 

99015 

00985 

39 

22 

96002 

03998 

97523 

02477 

99040 

00960 

38 

28 

96028 

03972 

97548 

02452 

99065 

00935 

37 

24 

96053 

03947 

97573 

02427 

99090 

00910 

36 

25 

96078 

03922 

97598 

0^402 

99116 

008S4 

35 

26 

96104 

03896 

97624 

02376 

99141 

00859 

34 

27 

96129 

03871 

97649 

02351 

99166 

00834 

33 

28 

96155 

03845 

97674 

02326 

99191 

00809 

82 

29 

96180 

03820 

97700 

02300 

99217 

00783 

31 

30 

9.96205 

10.03795 

9.97725 

10.02275 

9.99242 

10.00758 

30 

31 

96231 

03769 

97750 

02250 

99267 

00733 

29 

32 

96256 

03744 

97776 

02224 

99293 

00707 

28 

33 

96281 

03719 

97801 

02199 

99318 

00682 

27 

34 

96307 

03693 

97826 

02174 

99343 

00657 

26 

35 

96332 

03668 

97851 

02149 

99368 

00632 

25 

36 

96357 

03643 

97877 

02123 

59394 

00606 

24 

37 

96383 

03617 

97902 

02098 

99419 

00581 

23 

38 

96408 

0359-2 

97927 

02073 

99444 

00556 

22 

39 

96433 

03567 

97953 

0^047 

99469 

00531 

21 

40 

9.96459 

10.03541 

9.97978 

10.02022 

9.99495 

10.00505 

20 

41 

96484 

03516 

9b003 

01997 

99520 

00480 

19 

42 

96510 

03490 

98029 

01971 

99545 

00455 

18 

43 

96535 

03465 

98054 

C1946 

99570 

00430 

17 

44 

96560 

03440 

98079 

01921 

99596 

00404 

16 

45 

965S6 

03414 

98104 

01896 

99621 

00379 

15 

46 

96611 

03389 

98130 

01870 

99646 

00354 

14 

47 

96636 

03364 

98155 

01845 

99672 

00328 

13 

48 

96662 

03338 

98180 

01880 

99697 

00303 

12 

49 

96687 

03313 

98206 

01794 

99722 

00278 

11 

50 

9.96712 

10.03288 

9.98231 

10.01769 

9.99747 

10.00253 

10 

51 

96738 

03262 

98256 

01744 

99773 

00227 

9 

52 

96763 

03237 

98281 

01719 

99798 

00202 

8 

53 

96788 

03212 

98307 

01693 

99823 

00177 

7 

54 

96814 

03186 

98332 

01668 

99848 

00152 

6 

55 

96839 

03161 

98357 

01643 

99874 

00126 

5 

W 

96864 

03136 

98383 

01617 

99899 

00101 

4 

57 

96890 

03110 

98408 

01592 

99924 

00076 

3 

58 

06915 

03085 

98433 

01567 

99949 

00051 

2 

59 

96940 

03060 

98458 

01542 

99975 

00025 

1 

60 

96966 

03034 

98484 

01516 

10.00000 

00000 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

47° 

*«° 

45<>      I 

TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 


Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

t 

o 

2 
3 
4 

.00000 
.00029 
.00058 
.00087 
.00116 

One 
One 
One 
One 
One 

.00000 
.00029 
.00058 
.00087 
.00116 

Infinite 
3437-75 
1718.87 
1145.92 
809.436 

.01745 
.01774 
.01803 
.01832 
.01862 

.99985 
-99984 
.99984 
.99983 
.99983 

.01746 
.01775 
.01804 
.01833 
.01862 

57.2900 
56.3506 
55-4415 
54-5613 
53-7086 

60 

59 
58 
57 
56 

5 
6 

8 
g 

.00145 
.00175 
.00204 
.00233 
.00262 

One 
One 
One 
One 
One 

.00145 
.00175 
.00204 
.00233 

.00262 

687-549 
572-957 
491.106 
429.718 
381.971 

.01891 
.01920 
.01949 
.01978 
.02007 

.99982 
.99982 
.99981 
.99980 
.99980 

.01891 
.01920 
.01949 
.01978 
.02007 

52.8821 
52-0807 
51-3032 
50.5485 
49.8157 

55 
54 
53 
52 
51 

_£  — 

10 

11 
12 
13 
14 

.00291 
.00320 
.00349 
.00378 
.00407 

One 
.99999 
.99999 
.99999 

.99999 

.00291 
.00320 
.00349 
.00378 
.00407 

343-774 
312-521 
286.478 
264.441 
245.552 

.02036 
.02065 
.02094 
.02123 
.02152 

.99979 
.99979 
.99978 
.99977 
.99977 

.02036 
.02066 
.02095 
.02124 
.02153 

49*1039 
48-4121 
47.7395 
47.0853 
46.4489 

5O 

49 
48 

47 
46 

15 
16 
17 
18 
19 

.00436 

.00465 
.00495 
.00524 
.00553 

.99999 
.99999 
.99999 
.99999 
.99998 

.00436 
.00465 
.00495 
.00524 
.00553 

229-182 
214.858 
202-219 
190-984 
180-932 

.02131 
.02211 
.02240 
.02269 
.02298 

.99976 
-99976 
.99975 
-99974 
-99974 

.02182 
.02211 
.02240 
.02269 
.02298 

45-8294 
45-2261 
44-6386 
44-0661 
43-5081 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.00582 
.00611 
.00640 
.00669 
.00698 

.99998 
.99998 
.99998 
.99998 
.99998 

.00582 
.00611 
.00640 
.00669 
.00698 

171-885 
163-700 
156-259 
149-465 
143-237 

.02327 
.02356 
.02385 
.02414 
.02443 

.99973 
.99972 
.99972 
.99971 
.99970 

.02328 
.02357 
.02386 
.02415 
.02444 

42-9641 
42-4335 
41.9158 
41.4106 
40.9174 

4O 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.00727 
.00756 
$0785 
.00814 
.00844 

.99997 
.99997 
.99997 
.99997 
.99996 

.00727 
.00756 
.00785 
.00815 
.00844 

137.507 
132-219 
127-321 
122.774 
118.540 

.02472 
.02501 
.02530 
.02560 
.02589 

.99969 
.99969 
.99968 
.99967 
.99966 

.02473 
.02502 
.02531 
.02560 
.02589 

40.4358 
39-9655 
39.5059 
39.0568 
38-6177 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.00873 
.00902 
.00931 
.00960 
.00989 

.99998 
.99996 
.99996 
.99995 
.99995 

.00873 
.00902 
.00931 
.00960 

.00989 

114-589 
110.892 
107.426 
104.171 
101.107 

.02618 
.02647 
.02676 
.02705 
.02734 

.99966 
.99965 
.99964 
.99963 
-99963 

.02619 
.02648 
.02677 
.02706 
.02735 

38-1885 
37-7686 
37.3579 
36.9560 
36.5627 

30 

29 
28 
27 
26 

35 
36 
87 
38 

39 

.01018 
.01047 
.01076 
.01105 
.01134 

.99995 
.99995 
.99994 
.99994 
.99994 

.01018 
.01047 
.01076 
.01105 
.01135 

98-2179 
95.4895 
92-9085 
90.4633 
88-1436 

.02763 
.02792 
.02821 
.02850 
.02879 

.99962 
.99961 
.99960 
.99959 
.99959 

.02764 
-02793 
.02822 
.02851 
.02881 

36.1776 
35-8006 
35.4313 
35-C695 

34-7151 

25 
24 
23 
22 
-21 

40 
41 
42 
43 

44 

.01164 
.01193 

.01222- 
.01251 
.01280 

.99993 
.99993 
.99993 
.99992 
.99992 

.01164 
.01193 
.01222 
.01251 
.01280 

85-9398 
83-8435 
8L8470 
79.9434 
78.1263 

•02908 
•02938 
•02967 
•02996 
.03025 

-99958 
-99957 
-99956 
.99955 
.99954 

•02910 
-02939 
.02968 
.02997 
.03026 

34-3678 
34.0273 
33-6935 
33-3662 
33-0452 

20 

19 
18 
17 
.18 

45 
46 
47 
48 
49 

.01309 
.01338 
.01367 
.01396 
•01425 

.99991 
.99991 
.99991 
.99990 
.99990 

•01309 
.01338 
.01367 
.01396 
.01425 

76-3900 
74.7292 
73-1390 
71-6151 
70-1533 

•03054 
.03083 
•03112 
-03141 
.03170 

.99953 
.99952 
.99952 
.99951 
.99950 

.03055 
-03084 
.03114 
-03143 
.03172 

32-7303 
32-4213 
32-1181 
31-8205 
31-5234 

15 
14 
13 
12 
.11 

50 

51 
52 
53 
54 

.01454 
•01483 
•01513 
.01542 
.01571 

.99989 
.99989 
.99989 
.99988 
.99988 

.01455 
.01484 
.01513 
.01542 
.01571 

68.7501 
67.4019 
66.1055 
64-8580 
63.6567 

.03199 
-03228 
.03257 
.03286 
.03316 

.99949 
.99948 
.99947 
.99946 
.99945 

.03201 
.03230 
-03259 
.03288 
.03317 

31.2416 
30-9599 
30-6833 
30.4116 
30-1446 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

.01600 
•01629 
.01658 
•01687 
.01716 

.99987 
.99987 
.99986 
.99986 
.99985 

.01600 
.01629 
.01658 
.01687 
.01716 

62.4992 
61.3829 
60-3058 
59.2659 
58.2612 

.03345 
-03374 
.03403 
.03432 
-03461 

.99944 
.99943 
.99942 
.99941 
.99940 

-03346 
.03376 
.03405 
.03434 
-03463 

29.8823 
29.6245 
29.3711 
29-1220 
28.8771 

5 
4 
3 
2 
1 

60 

.01745 

.99985 

.01746 

57-2900 

-03490 

.99939 

.03492 

28-6363 

-° 

/ 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

1 

19° 

2( 

)6 

€ 

18° 

Inserted  by  permission  from  Webb's  Railroad  Construction. 


TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
2°  3° 


1 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

t 

0 

2 
3 

03490 
03519 
03548 
03577 
03606 

99939 
.99938 
.99937 
.99936 
.99935 

03492 
03521 
03550 
03579 
03609 

28-6363 
28-3994 
28.1664 
27-9372 
27-7117 

05234 
05263 
05292 
05321 
05350 

99863 
99861 
99860 
99858 
99857 

05241 
05270 
05299 
05328 
05357 

19-0511 
18-9755 
18-8711 
18-7678 
18.6656 

60 

59 
58 
57 
58 

~5~ 
6 
7 
8 
g 

03635 
03664 
03693 
03723 
03752 

.99934 
.99933 
.99932 
.99931 
.99930 

03638 
03667 
03696 
03725 
03754 

27-4899 
27-2715 
27-0566 
26.8450 
26-6367 

05379 
05408 
05437 
05466 
05495 

99855 
99854 
99852 
99851 
99849 

.05387 
.05416 
.05445 
.05474 
.05503 

18-5645 
18-4645 
18.3655 
18-2677 
18-1708 

55 
54 
53 
52 
_51 

10 

11 
12 
13 
14 

03781 
03810 
03839 
03868 
03897 

.99929 
.99927 
99926 
99925 
99924 

03783 
03812 
03842 
03871 
03900 

26.4316 
26-2296 
26-0307 
25.8348 
25-6418 

05524 
05553 
05582 
05611 
05640 

99847 
99846 
99844 
99842 
99841 

.05533 
.05562 
.05591 
.05620 
.05649 

18-0750 
17-9802 
17.8863 
17-7934 
17-7015 

50 

49 
48 
47 
48 

15~ 
16 
17 
18 
19 

03926 
03955 
03984 
04013 
04042 

99923 
99922 
99921 
99919 
99918 

03929 
03958 
03987 
04016 
04046 

25.4517 
25-2644 
25.0798 
24-8978 
24-7185 

05669 
05698 
.05727 
.05756 
.05785 

99839 
99838 
99836 
99834 
99ff33 

.05678 
.05708 
.05737 
.05766 
.05795 

17.6108 
17.5205 
17-4314 
17-3432 

17-2558 

45 
44 
43 
42 
-M 

20 

21 
22 
23 
24 

04071 
.04100 
.04129 
.04159 
.04188 

99917 
99916 
99915 
99913 
99912 

04075 
04104 
04133 
04162 
04191 

24-5418 
24.3675 
24.1957 
24-0263 
23-8593 

.05814 
.05844 
.05873 
.05902 
.05931 

99831 
99829 
99827 
.99826 
.99824 

.05824 
.05854 
.05883 
.05912 
-05941 

17-1693 
17-0837 
16-9990 
16-9150 
16-8319 

40 

39 
38 
37 
_38 

25~ 
28 
27 
28 
29 

.04217 
.04246 
.04275 
.04304 
.04333 

99911 
99910 
.99909 
.99907 
.99906 

04220 
04250 
04279 
04308 
04337 

23.6945 
23-5321 
23-3718 
23-2137 
23-0577 

.05960 
.05989 
.06018 
.06047 
.06076 

.99822 
.99821 
.99819 
.99817 
.99815 

.05970 
.05999 
.06029 
.06058 
.06087 

16.7498 
16-6681 
16-5874 
16-5075 
16-4283 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.04362 
.04391 
.04420 
.04449 
.04478 

.99905 
.99904 
.99902 
.99901 
.99900 

04366 
.04395 
.04424 
.04454 
.04483 

22-9038 
22.7519 
22-6020 
22-4541 
22-3081 

.06105 
.06134 
.06163 
.06192 
.06221 

.99813 
.99812 
.99810 
.99808 
.99806 

.06116 
.06145 
-06175 
-06204 
.06233 

16-3499 
16-2722 
16-1952 
18.1190 
16-0435 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.04507 
.04536 
.04565 
.04594 
.04623 

.99898 
.99897 
.99896 
.99894 
.99893 

.04512 
.04541 
.04570 
.04599 
.04628 

22-1640 
22-0217 
21-8813 
21-7426 
21-6056 

.06250 
.06279 
.06308 
.06337 
.06366 

.99804 
.99803 
.99801 
.99799 
.99797 

-06262 
.06291 
.06321 
.06350 

.06379 

15-9687 
15-8945 
15-8211 
15-7483 

15-6762 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

.04653 
.04682 
.04711 
.04740 
.04769 

.99892 
.99890 
.99889 
.99888 
,99886 

.04658 
.04687 
.04716 
.04745 
•04774 

21-4704 
21-3369 
21-2049 
21-0747 
20-9460 

.06395 
.06424 
.06453 
.06482 
.06511 

.99795 
.99793 
.99792 
.99790 
.99788 

.06408 
.06437 
.06467 
.06496 
.06525 

15-6048 
15-5340 
15-4638 
15-3943 
15-3254 

20 

19 
18 
17 
18 

45 
46 
47 
48 
49 

.04798 
•04827 
.04856 
.04885 
.04914 

.99885 
.99883 
.99882 
•99881 
.99879 

.04803 
.04833 
04862 
•04891 
.04920 

20-8188 
20-6932 
20-5691 
20-4465 
20-3253 

.06540 
.06569 
.06598 
.06627 
•06656 

.99786 
.99784 
.99782 
.99780 
.99778 

.06554 
.06584 
.06613 
.06642 
.06671 

15-2571 
15-1893 
15-1222 
15-0557 
14-9898 

Ib 
14 
13 
12 
11 

50 

51 
52 
53 
54 

.04943 
.04972 
.05001 
.05030 
.05059 

.99878 
.99876 
.99875 
.99873 
.99872 

.04949 
.04978 
.05007 
.05037 
.05066 

20-2056 
20-0872 
19-9702 
19-8546 
19-7403 

.06685 
-06714 
-06743 
.06773 
.06802 

.99776 
.99774 
.99772 
.99770 
.99768 

.06700 
.06730 
.06759 
.06788 
.06817 

14-9244 
14-8596 
14-7954 
14.7317 
14-6685 

1O 

9 
8 
7 
8 

55 
56 
67 
68 

59 

.05088 
.05117 
.05146 
.05175 
.05205 

.99870 
.99869 
.99867 
.99866 
.99864 

.05095 
.05124 
.05153 
.05182 
.05212 

19-6273 
19-5156 
19-4051 
19-2959 
19-1879 

.06831 
.06860 
.06889 
.06918 
.06947 

.99766 
.99764 
.99762 
.99760 
.99758 

.06847 
.06876 
.06905 
.06934 
.06963_ 

14.6059 
14-5438 
14-4823 
14-4212 
14-3607 

5 

3 

2 

60 

.05234 

.99863 

.05241 

19-0811 

.06976 

.99756 

.06993 

14-3007 

0 

/ 

Cos. 

Sin. 

Cot. 

an. 

Cos. 

Sin. 

Cot. 

Tan. 

* 

>7° 

207 

86° 

TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS.  AND  COTANGENT& 

4°  5° 


1 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

' 

0 

2 
3 

4 

.06976 
.07005 
.07034 
.07063 
.07092 

.99756 
.99754 
.99752 
.99750 
.99748 

.06993 
.07022 
.07051 
.07080 
.07110 

14.3007 
14.2411 
14-1821 
14.1235 
14.0655 

.08716 
.08745 
.08774 
.08803 
.08831 

99619 
.99617 
.99614 
-99612 
-99609 

08749 
08778 
08807 
.08837 
.08866 

11.4301 
11.3919 
11.3540 
11.3163 
11.2789 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.07121 
.07150 
.07179 
.07208 
.07237 

.99746 
.99744 
.99742 
.99740 
.99738 

.07139 
.07168 
.07197 
.07227 
.07256 

14-0079 
13.9507 
13.8940 
13-8378 
13.7821 

.08860 
.08889 
.08918 
.08947 
.08976 

-99607 
-99604 
.99602 
.99599 
-99596 

.08895 
-08925 
-08954 
.08983 
.09013 

11.2417 
11.2048 
11.1681 
11.1316 
11.0954 

55 
54 
53 
52 
51 

10 

11 
12 
13 
,14 

.07266 
.07295 
.07324 
.07353 
.07382 

.99736 
.99734 
.99731 
.99729 

.99727 

.07285 
.07314 
.07344 
.07373 

.07402 

13.7267 
13-6719 
13.6174 
13.5634 
13.5098 

.09005 
.09034 
.09063 
.09092 
.09121 

.99594 
.99591 
.99588 
-99586 
.99583 

-09042 
.09071 
-09101 
.09130 
.09159 

11.0594 
11.0237 
10-9882 
10-9529 
10.9178 

50 

49 
48 
47 
4B 

15 
16 
17 
18 
19 

.07411 
.07440 
.07469 
.07498 
.07527 

.99725 
.99723 
.99721 
.99719 
.99716 

.07431 
.07461 
.07490 
.07519 
.07548 

13-4566 
13.4039 
13.3515 
13-2996 
13-2480 

-09150 
.09179 
.09208 

.09237 
.09266 

.99580 
.99578 
.9C575 
.9&S72 
.99570 

.09189 
.09218 
.09247 
.09277 
.09306 

10-8829 
10-8483 
10-8139 
10-7797 
10-7457 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.07556 
.07585 
.07614 
.07643 
.07672 

.99714 
.99712 
.99710 
.99708 

.99705 

.07578 

.07607 
.07636 
.07665 
.07695 

13.1969 
13-1461 
13-0958 
13.0458 
12-9962 

.09295 
.09324 
.09353 
.09382 
.09411 

.99567 
.99564 
.99562 
.99559 
.99556 

.09335 

.09365 
.09394 
-09423 
-09453 

10-7119 
10.6783 
10.6450 
10-6118 
10-5789 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

•07701 
.07730 
.07759 
.07788 
.07817 

.99703 
.99701 
.99699 
.99696 
.99694 

.07724 
.07753 
.07782 
.07812 
.07841 

12-9469 
12-8981 
12.8496 
12.8014 
12.7536 

.09440 
-09469 
-09498 
.09527 
.09556 

.99553 
.99551 
.99548 
.99545 
.99542 

.09482 
.09511 
.09541 
.09570 
.09600 

10-5462 
10-5136 
10.4813 
10-4491 
10-4172 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.07846 
.07875 
.07904 
.07933 
.07962 

.99692 
.99689 
.99687 
.99685 
.99683 

.07870 
.07899 
.07929 
.07958 
.07987 

12.7062 
12.6591 
12-6124 
12.5660 
12.5199 

•09585 
.09614 
.09642 
.09671 
.09700 

.99540 
-99537 
-99534 
-99531 
-99528 

-09629 
.09658 
.09688 
.09717 
.09746 

10-3854 
10-3538 
10-3224 
10.2913 
10.2602 

30 

29 
28 
27 
26 

35 
36 
87 
38 
89 

.07991 
.08020 
.08049 
.08078 
.08107 

.99680 
.99678 
.99676 
.99673 
.99671 

.08017 
.08046 
.08075 
.08104 
.08134 

12.4742 
12.4288 
12-3838 
12.3390 

12-2946 

.09729 
-09758 
.09787 
.09816 
.09845 

-99526 
.99523 
.99520 
.99517 
.99514 

-09776 
.09805 
•09834 
.09864 
.09893 

10.2294 
10-1988 
10.1683 
i0.1381 
10.1080 

25 
24 
23 
22 
,.21 

40 

41 
42 
43 
44 

.08136 
.08165 
.08194 
.08223 
.08252 

.99668 
.99666 
.99664 
.99661 
.99659 

•08163 
.08192 
.08221 
.08251 
.08280 

12-2505 
12.2067 
12.1632 
12.1201 
12.0772 

.09874 
.09903 
.09932 
•09961 
.09990 

•99511 
•99508 
.99506 
.99503 
.99500 

.09923 
.09952 
.09981 
.10011 
.10040 

10-0780 
10-0483 
10-0187 
9-98931 
9.96007 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

•08281 
.08310 
.08339 
.08368 
.08397 

.99657 
.99654 
.99652 
.99649 
.99647 

.08309 
.08339 
.08368 
.08397 
.08427 

12.0346 
11.9923 
11.9504 
11-9087 
11-8673 

-10019 
.10048 
-10077 
.10106 
.10135 

.99497 
.99494 
.99491 
.99488 
.99485 

.10069 
.10099 
.10128 
.10158 
-10187 

9-93101 
9-90211 
9-87338 
9  •  84482 
9.81641 

15 
14 
13 
12 
_!! 

50 

51 
52 
53 
54 

.08426 
.08455 
.08484 
.08513 
.08542 

.99644 
.99642 
.99639 
.99637 
.99635 

.08456 
.08485 
.08514 
.08544 
.08573 

11.8262 
11.7853 
11.7448 
11.7045 
11.6645 

-10164 
-10192 
-10221 
.10250 
.10279 

-99482 
-99479 
.99476 
.99473 
.99470 

.10216 
-10246 
-10275 
.10305 
.10334 

9-78817 
9-76009 
9-73217 
9.70441 
9-67680 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

.08571 
.08600 
.08629 
.08658 
.08687 

.99632 
.99630 
.99627 
.99625 
.99622 

.08602 
.08632 
.08661 
.08690 
.08720 

11.6248 
11.5853 
11.5461 
11.5072 
11.4685 

.10308 
.10337 
.10366 
.10395 
.10424 

.99467 
-99464 
.99461 
-99458 
-99455 

.10363 
.10393 
.10422 
•10452 
.10481 

9.64935 
9-62205 
9.59490 
9-56791 
9-54106 

5 
4 
3 
2 

60 

.08716 

.99619 

.08749 

11-4301 

.10453 

•99452 

.10510 

9.51436 

O 

1  / 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

86° 


208 


84° 


TABLE  10  —NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
6°  7° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

t 

o 

1 

2 
3 

10453 
10482 
10511 
10540 
10569 

.99452 
.99449 
.99446 
.99443 
.99440 

10510 
10540 

10569 
10599 
10628 

9.51436 
9.48781 
9.46141 
9.43515 
9.40904 

12187 
12216 
12245 
12274 
12302 

99255 
99251 
99248 
99244 
99240 

12278 
12308 
.12338 
.12367 
.12397 

8.14435 
8.12481 
8.10536 
8.08600 
8.06674 

60 

59 
58 
57 
56 

5 
6 
7 

8 
9 

10597 
10626 
10655 
.10684 

.10713 

.99437 
.99434 
.99431 
.99428 
.99424 

10657 
.10687 
.10716 
.10746 
.10775 

9.38307 
9.35724 
9.33155 
9.30599 
9.28058 

.12331 

.12360 
•12389 
.12418 
.12447 

99237 
.99233 
.99230 
.99226 

•99222 

•12426 
•12456 
.12485 
.12515 
.12544 

8-04756 
8.02848 
8.00948 
7.99058 
7-97176 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.10742 
.10771 
.10800 
.10829 
.10858 

.99421 
.99418 
.99415 
.99412 
.99409 

.10805 
.10834 
.10863 
.10893 
.10922 

9-25530 
9.23016 
9-20516 
9.18028 
9-15554 

.12476 
.12504 
.12533 
.12562 
.12591 

.99219 
•99215 
.99211 
.99208 
.99204 

.12574 
.12603 
.12633 
.12662 
.12692 

7.95302 
7-93438 
7-91582 
7-89734 
7-87895 

50 

49 
48 
47 
46 

15 
16 
17 
18 

19 

.10887 
.10916 

.10945 
.10973 
.11002 

.99406 
.99402 
.99399 
.99396 
.99393 

.10952 
.10981 
.11011 
.11040 
.11070 

9.13093 
9-10646 
9.08211 
9.05789 
9-03379 

.12620 
.12649 
.12678 
.12706 
.12735 

.99200 
.99197 
.99193 
.99189 
.99186 

.12722 
.12751 
.12781 
.12810 
.12840 

7-86064 
7-84242 
7-82428 
7-80622 
7.78825 

45 
44 
43 
42 
-il 

2O 

21 
22 
23 
24 

.11031 
.11060 
.11089 
-11118 
.11147 

.99390 
.99386 
.99383 
.99380 
.99377 

.11099 
.11128 
.11158 
.11187 
.11217 

9.00983 
8-98598 
8-96227 
8.93867 
8.91520 

.12764 
.12793 
.12822 
.12851 
.12880 

.99182 
.99178 
•99175 
.99171 
.99167 

.12869 
.12899 
.12929 
.12958 
.12988 

7-77035 
7-75254 
7-73480 
7.71715 
7.69957 

40 

39 
38 
37 
36 

25 
26 
£7 
28 
29 

.11176 

•11205 
11234 
.11263 
.11291 

.99374 
.99370 
.99367 
.99364 
.99360 

-11246 
•11276 
.11305 
.11335 
.11364 

8-89185 

8-86862 
8.84551 
8-82252 
8-79964 

.12908 
.12937 
.12966 
.12995 
.13024 

•99163 
-99160 
•99156 
.99152 
.99148 

.13017 
.13047 
.13076 
.13106 
.13136 

7.68208 
7.66466 
7.64732 
7.63005 
7.61287 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.11320 
•11349 
.11378 
•11407 
.11436 

.99357 

.99354 
.99351 
-99347 
.99344 

.11394 
.11423 
.11452 
.11482 
.11511 

8-77689 
8-75425 
8-73172 
8-70931 
8.68701 

.13053 
.13081 
.13110 
.13139 
.13168 

.99144 
.99141 
.99137 
.99133 
.99129 

.13165 
.13195 
.13224 
.13254 
.13284 

7-59575 
7-57872 
7-56176 
7-54487 
7-52806 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.11465 
.11494 
.11523 
.11552 
•11580 

.99341 
.99337 
.99334 
.99331 
.99327 

.11541 
.11570 
.11600 
.11629 
.11659 

8.66482 
8-64275 
8.-  62078 
8.59893 
8.57718 

•13197 
.13226 
.13254 
.13283 
.13312 

.99125 
.99122 
.99118 
.99114 
•99110 

.13313 
.13343 
.13372 
.13402 
.13432 

7.51132 
7-49465 
7-47806 
7-46154 
7-44509 

25 
24 
23 
22 
21 

40 

41 
42 
43 

44 

.11609 
•11638 
11667 
.11698 
.11725 

.99324 
.99320 
.99317 
.99314 
.99310 

.11688 
.11718 
.11747 
.11777 
.11806 

8-55555 
8-53402 
8.51259 
8-49128 
8-47007 

•13341 
.13370 
.13399 
.13427 
.13456 

.99106 
.90102 
.99098 
.99094 
.99091 

.13461 
.13491 
.13521 
.13550 

.13580 

7-42871 
7-41240 
7-39616 
7-37999 
7-36389 

20 

19 
18 
17 
18 

45 
46 
47 
48 
49 

.11754 
.11783 
.11812 
•11840 
.11869 

.99307 
.99303 
.99300 
•99297 
.99293 

.11836 
.11865 
.11895 
.11924 
•11954 

8-44896 
8-42795 
8-40705 
8-38625 
8-36555 

.13485 
•13514 
.13543 
.13572 
.13600 

.99087 
.99083 
.99079 
.99075 
.99071 

.13609 
.13639 
.13669 
.13698 
.13728 

7-34786 
7-33190 
7-31600 
7-30018 
7-28442 

15 
14 
13 
12 
11 

50 

51 
52 
53 

54 

.11898 
.11927 
.11956 
.11985 
.12014 

.99290 
.99286 
.99283 
•99279 
.99276 

.11983 
=12013 
.12042 
•12072 
.12101 

8.34496 
8.32446 
8-30406 
8-28376 
8.26355 

.13629 
.13658 
•13687 
.13716 
.13744 

.99067 
.99063 
.99059 
.99055 
.99051 

.13758 
•13787 
.13817 
.13846 
.13876 

7-26873 
7-25310 
7-23754 
7-22204 
7-20661 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•12043 
•12071 
.12100 
.12129 
.12158 

.99272 
.99269 
.99265 
•99262 
.99258 

.12131 
.12160 
.12190 
•12219 
•12249 

8.24345 
8-22344 
8.20352 
8.18370 
8.16398 

.13773 
.13802 
.13831 
.13860 
.13889 

.99047 
.99043 
.99039 
•99035 
.99031 

.13906 
.13935 
.13965 
.13995 
.14024 

7-19125 
7-17594 
7-16071 
7-14553 
7-13042 

5 

3 
2 

60 

.12187 

.99255 

.12278 

8.14435 

.13917 

.99027 

.14054 

7-11537 

0 

1 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

/ 

S3? 


209 


TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
8°  9° 


t 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

t 

60 

59 
58 
57 
56 
55 
54 
53 
52 
51 

0 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
LL 
15 
16 
17 
18 
!2_ 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

.13917 

•13946 
•13975 
•14004 
.14033 

•99027 
.99023 
.99019 
.99015 
.99011 

.14054 
.14084 
.14113 
.14143 
.14173 

7.11537 
7-10038 
7-08546 
7-07059 
7-05579 

.15643 
.15672 
.15701 
.15730 
.15758 

.98769 
.98764 
.98760 
.98755 
.98751 

.15838 
.15868 
.15898 
.15928 
.15958 

6.31375 
6.30189 
6-29007 
6.27829 
6.26655 

.14061 
.14090 
.14119 
.14148 
.14177 

.99006 
.99002 
.98998 
.98994 
.98990 

.14202 
.14232 
.14262 
.14291 
.14321 

7-04105 
7-02637 
7.01174 
6-99718 
6-98268 

-15787 
•15816 
.15845 
•15873 
•15902 

.98746 
•98741 
.98737 
•98732 
.98728 

.15988 
.16017 
.16047 
.16077 
.16107 

6-25488 
6.24321 
6-23160 
6-22003 
6-20851 

.14205 
.14234 
.14263 
.14292 
.14320 

.98986 
.98982 
.98978 
.98973 
.98969 

.14351 
.14381 
.14410 
.  14440 
.14470 

6-96823 
6-95385 
6-93952 
6-92525 
6-91104 

-15931 
.15959 
.15988 
.16017 
-16046 

•98723 
.98718 
.98714 
.98709 
.98704 

.16137 
.16167 
.16196 
.16226 
.16256 

6.19703 
6-18559 
6.17419 
6.16283 
6-15151 

5O 

49 
48 
47 
_46 
45 
44 
43 
42 
-il 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

.14349 
.14378 
.14407 
.14436 
.  14464 

.98965 
.98961 
.98957 
.98953 
.98948 

.14499 
.14529 
.14559 
.14588 
.14618 

6.89688 
6-88278 
6-86874 
6-85475 

6-84082 

.16074 
.16103 
.16132 
.16160 
.16189 

.98700 
•98695 
-98690 
-98686 
.98681 

.16286 
-16316 
.16346 
-16376 
-16405 

6.14023 
6.12899 
6.11779 
6.10664 
6-09552 

.14493 
•14522 
.14551 
.14580 

.14608 

.98944 
.98940 
.98936 
.98931 
.98927 

.14648 
.14678 
.14707 
.14737 
.14767 

6.82694 
6-81312 
6-79936 
6-78564 
6-77199 

.16218 
.16246 
.16275 
.16304 
.16333 

.98676 
.98671 
.98667 
.98662 
.98657 

.16435 
.16465 
.16495 
.16525 
.16555 

6.08444 
6.07340 
6-06240 
6.05143 
6-04051 

.14637 
.14666 
.14695 
.14723 

.14752 

.98923 
.98919 
.98914 
.98910 
.98906 

.14796 
.14826 
.14856 
.14886 
.14915 

6.75838 
6-74483 
6-73133 
6-71789 
6-70450 

.16361 
.16390 
.16419 
.16447 
.16476 

.98652 
.98648 
-98643 
.98638 
-98633 

.16585 
.16615 
.16645 
.16674 
.16704 

6.02962 
6.01878 
6-00797 
5.99720 
5.98646 

30 

31 
32 
33 
34_ 
35 
36 
37 
38 
39_ 
40 
41 
42 
43 
44 

.14781 
.14810 
.14838 
.14867 
.14896 

.98902 
.98897 
.98893 
.98889 
.98884 

.14945 
.14975 
.15005 
.15034 
.15064 

6.69116 
6-67787 
6-66463 
6-65144 
6.63831 

.16505 
.16533 
.16562 
.16591 
.16620 

-98629 
.98624 
.98619 
.98614 
.98609 

.16734 
-16764 
.16794 
-16824 
.16854 

5-97576 
5-96510 
5.95448 
5.94390 
5-93335 

30 

29 
28 
27 
26 
25 
24 
23 
22 
21 

.14925 
.14954 
.14982 
.15011 
.15040 

.98880 
.98876 
.98871 
.98867 
.98863 

.15094 
•15124 
.15153 
.15183 
.15213 

6-62523 
6.61219 
6-59921 
6-58627 
6.57339 

.16648 
.16677 
.16706 
-16734 
-16763 

.98604 
.98600 
.98595 
.98590 
.98585 

.16884 
.16914 
.16944 
.16974 
.17004 

5.92283 
5.91236 
5.90191 
5.89151 
5.88114 

.15069 
•15097 
.15126 
.15155 
.15184 

.98858 
.98854 
.98849 
.98845 
.98841 

•15243 
.15272 
.15302 
.15332 
.15362 

6.56055 
6.54777 
6.53503 
6.52234 
6.50970 

.16792 
.16820 
.16849 
.16878 

-16906 

.98580 
.98575 
.98570 
.98565 
-98561 

.17033 
.17063 
.17093 
-17123 
.17153 

5.87080 

5-86051 
5-85024 
5-84001 
5.82982 

20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 

3 
2 

-0 

45 
46 
47 
48 
49 

.15212 
.15241 
.15270 
.15299 
•15327 

.98836 
.98832 
.98827 
.98823 
.98818 

.15391 
.15421 
.15451 
.15481 
.15511 

6.49710 
6.48456 
6-47206 
6.45961 
6.44720 

.16935 
-16964 
•16992 
•17021 
.17050 

.98556 
-98551 
-98548 
-98541 
.98536 

.17183 
.17213 
.17243 
.17273 
.17303 

5.81966 
5.80953 
5.79944 
5-78938 
5-77936 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

•15356 
.15385 
.15414 
.15442 
.15471 

•98814 
.98809 
.98905 
•98800 
•98796 

•15540 
.15570 
.15600 
.15630 
.15660 

6-43484 
6.42253 
6.41026 
6.39804 
6-38587 

•17078 
.17107 
.17136 
.17164 
•17193 

.98531 
-98526 
-98521 
.98516 
.98511 

.17333 
.17363 
.17393 

.17423 
.17453 

5-76937 
5.75941 
5.74949 
5-73960 
5.72974 

•15500 
-15529 
•15557 
•15586 
.15615 

•98791 
.98787 
.98782 
.98778 
.98773 

•15689 
.15719 
.15749 
.15779 
.15809 

6-37374 
6-36165 
6-34961 
6.33761 
6.32566 

.17222 
•17250 
•17279 
•17308 
-17336 

-98506 
.98501 
.98496 
.98491 
.98486 

-17483 
•17513 
-17543 
.17573 
.17603 

5-71992 
5-71013 
5-70037 
5-69064 
5-68094 

60. 

•15643 

-98769 

.15838 

6.31375 

.17365 

.98481 

.17633 

5-67128 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

81° 


210 


80° 


-NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
10°  11° 


t 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

2 
3 

~5~ 
8 
7 
8 

JL_ 

10 

11 

12 
13 
14 
15 
16 
17 
18 
12- 
20 
21 
22 
23 
24^ 

25 
26 
27 
28 
29_ 
30 
31 
32 
33 

3JL 

35 
36 
37 
38 
82- 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

.17365 
•17393 

-17422 
•17451 
.17479 

.98481 

.98476 
.98471 
.98466 
.98461 

•17633 
-17663 
•17693 
.17723 
-17753 

5-67128 
5-66165 
5-65205 
5-64248 
5-63295 

-19081 
-19109 
-19138 
•19167 
•19195 

.'98163 
-98157 
-98152 
-98146 
-98140 

-19438 
-19468 
.19498 
•19529 
.19559 

5.14455 
5-13658 
5-12862 
5-12069 
5-11279 

60 

59 
58 
57 
56 

•17508 
•17537 
.17565 
.17594 
-17623 

.98455 
.98450 
.98445 
.98440 
.98435 

.17783 
.17813 
.17843 
.17873 
•17903 

5-62344 
5-61397 
5  .  60452 
5-59511 
5.58573 

•19224 
•  19.252 
-19281 
-19309 
-19338 

•98135 
-98i29 
-98124 
-98118 
98112 

•19589 
•19619 
•19649 
.19680 
.19710 

5-10490 
5.09704 
5.08921 
5-08139 
5-07360 

55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 

.17651 
•17680 
.17708 
.17737 
•17766 

.98430 
.98425 
•98420 
.98414 
.98409 

17933 
.17963 
.17993 
•18023 
.18053 

5-57638 
5.56706 
5-65777 
5.54851 
5-53927 

.19366 
-19395 
.19423 
•19452 
-19481 

-98UD7 
-98101 
-98096 
-98090 
-98084 

.19740 
•19770 
.19801 
-19831 
.19861 

5-06584 
5-05809 
5-05037 
5-04267 
5-03495 

.17794 
.17823 
.17852 
.17880 
.17909 

.98404 
.98399 
.98394 
.98389 
.98383 

•18083 
.18113 
.  18143 
•18173 
.18203 

5-53007 
5-52090 
5-51176 
5-50264 
5-49356 

.19509 
•19538 
-19566 
-19595 

-19623 

•98079 
.98073 
.98067 
.98061 
.98056 

•19891 
•19921 
•19852 
.19982 
-20012 

5.02734 
5-01971 
5-01210 
5-00451 

4-99695 

17937 
.17966 
.17995 
.18023 
.18052 

.98378 
.98373 
.98368 
.98362 
.98357 

.18233 
.18263 
.18293 
.18323 
.18353 
•18384 
.18414 
.18444 
.18474 
.18504 

5-48451 
5-47548 
5-46648 
5-45751 
5-44857 

.19652 
-19680 
•19709 
•19737 
-19766 

.98050 
.98044 
.98039 
.98033 

.98027 

.  20042 
-20073 
-20103 
-20133 
-20164 

4-98940 
4-98188 
4-97438 
4.96690 
4-95945 

.18081 
.18109 
-18138 
.18166 
.18195 

.98352 
.98347 
.98341 
.98336 
.98331 

5-43966 
5-43077 
5-42192 
5-41309 
5-40429 

.19794 
-19823 
-19851 
.19880 
-19908 

.98021 
.98016 
.98010 
•98004 
-97998 

•20194 
.20224 
•20254 
•20285 
-20315 

4-95201 
4.94460 
4-93721 
4-92984 
4-92249 

•18224 
.18252 
.18281 
.18309 
.18338 

.98325 
.98320 
.98315 
.98310 
.98304 

•18534 
.18564 
.18594 
•18624 
.18654 

5-39552 
5-38677 
5.37805 
5-36936 
5-36070 

-19937 
-19965 
-19994 
-20022 
-20051 

.97992 
-97987 
-97981 
.97975 
-97969 

-  20345 
-20376 
-20406 
-20436 
•20466 

4-91516 
4-90785 
4-90056 
4-89330 
4-88605 

3O 

29 
28 
27 
26 
25 
24 
23 
22 
21 

.18367 
.18395 
.18424 
•18452 
.18481 

.98299 
.98294 
•98288 
.982b3 
.98277 

.18634 
•18714 
.18745 
.18775 
.18805 

5-35206 
5.34345 
5-33487 
5-32631 
5  31778 

.20079 
-20108 
-20136 
-20165 
•20193 

-97963 
-97958 
.97952 
.97946 
.97940 

.20497 
.20527 
•20557 
-20588 
-20618 

4-87882 
4-87162 
4-86444 
4-85727 
4  85013 

•18509 
•18538 
.18567 
.18595 
.18624 

.98272 
.98267 
.98261 
•98256 
•98250 

.18835 
.18865 
.18895 
.18925 
.18955 

5-30928 
5-30080 
5-29235 
5-28393 
5.27553 

-20222 
•20250 
-20279 
-20307 
-20336 

•97934 
.97928 
-97922 
.97916 
-97910 

-20648 
-20679 
-20709 
-20739 
-20770 

4-84300 
4-83590 
4-82882 
4-82175 
4-81471 

20 

19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 
5 

3 
2 

~~0 

.18652 
•18681 
•18710 
•18738 
.18767 

.98245 
•98240 
•98234 
•98229 
.98223 

.18986 
-19016 
.19046 
.19076 
.19106 

5-26715 
5-25880 
5.25C48 
5-24218 
5.23391 

-20364 
•20393 
-20421 
•20450 
•20478 

-97905 
.97899 
.97893 
-97887 
-97881 

.20800 
-20830 
-20861 
-20891 
-20921 

4-80769 
4-80068 
4-79370 
4-78673 
4-77978 

-18795 
.18824 
.18852 
.18881 
•18910 

.98218 
•98212 
.98207 
•98201 
.98196 

.19136 
-19166 
•19197 
.19227 
.19257 

5-22566 
5.21744 
5-20925 
5.20107 
5-19293 

-20507 
-20535 
-20563 
-20592 
-20620 

-97875 
-97869 
-97863 
-97857 
.97851 

•20952 
-20982 
-21013 
-21043 
-21073 

4.77286 
4-76595 
4-75906 
4-75219 
4-74534 

55 
56 
57 
58 
59 
60_ 

.18938 
.18967 
•18995 
.19024 
.19052 

.98190 
.98185 
.98179 
.98174 
•98168 

.19287 
•19317 
.19347 
•19378 
.19408 

5-18480 
5.17671 
5-16863 
5-16058 
5-15256 

-20649 
-206.77 
-20706 
-20734 
•20763 

-97845 
-97839 
.97833 
-97827 
.97821 

-21104 
-21134 
-21164 
-21195 
.21225 

4.73851 
4-73170 
4-72490 
4-71813 
4-71137 

-19081 

.98163 

•19438 

5-14455 

-20791 

-97815 

-21256 

4-70463 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

79° 


211 


78° 


TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
12°  13° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

1 

2 
3 
4 

.20791 
.20820 
•20848 
.20877 
.  20905 

.97815 
.97809 
•97803 
.97797 
•97791 

-21256 
-21286 
-21316 
.21347 
•21377 

4.70463 
4-69791 
4.69121 
4.68452 
4-67786 

-22495 
-22523 
-22552 
-22580 
-22608 

•97437 
-97430 
-97424 
-97417 
-97411 

-23087 
-23117 
-23148 
-23179 
•23209 

4-33148 
4-32573 
4-32001 
4-31430 
4-30860 

60 

59 
58 
57 
5B 

5 
6 
7 
8 
9 

•20933 
.20962 
.  2099-0 
.21019 
.21047 

.97784 
•97778 
.97772 
.97766 
.97760 

•21408 
•21438 
•21469 
-21499 
.21529 

4-67121 
4.66458 
4-65797 
4-65138 
4-64480 

-22637 
-22665 
-22693 
-22722 
.22750 

-97404 
.97398 
•97391 
•97384 
-97378 

-23240 
-23271 
-23301 
•23332 
•23363 

4-30291 
4-29724 
4-29159 
4-28595 
4-28032 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.21076 
.21104 
-2U32 
.21161 
.21189 

.97754 
•97748 
.97742 
•97735 
.97729 

-21560 
•21590 
-21621 
•21651 
-21682 

4-63825 
4-63171 
4-62518 
4-61868 
4-61219 

.22778 
.22807 
-22835 
-22863 
-22892 

•97371 
•97365 
.97358 
.97351 
-97345 

.23393 
-23424 
-23455 
-23485 
•23516 

4-27471 
4-26911 
4-26352 
4-25795 
4-25239 

50 

49 
48 
47 
4B 

15 
16 
17 
18 

19 

.21218 

•2124S 
-21275 
•21303 
-21331 

.97723 
•97717 
.97711 
•97705 
•97698 

.21712 
.21743 
.21773 
-21804 
-21834 

4-60572 
4-59927 
4-59283 
4-58641 
4-58001 

•22920 
-22948 
-22977 
-23005 

.23033 

.97338 
.97331 
•97325 
.97318 
-97311 

-23547 
-23578 
.23608 
-23639 
-23670 

4-24685 
4.24132 
4-23580 
4.23030 
4-22481 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.21360 
.21388 
.21417 
.  21445 
•21474 

.97692 
.97686 
.97680 
-97673 
•97667 

.21864 
-21895 
-21925 
-21956 
•21986 

4-57363 
4-56726 
4.56091 
4.55458 
4-54826 

-23062 
-23090 
-23118 
-23146 
.23175 

•97304 
-97298 
-97291 
-97284 
.97278 

.23700 
-23731 
-23762 
-23793 
-23823 

4-21933 
4-21387 
4-20842 
4.20298 
4-19756 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

•21502 
•21530 
•21559 
•21587 
.21616 

.97661 
.97655 
•97648 
•97642 
-97636 

.22017 
•22047 
•22078 
•22108 
.22139 

4.54196 
4.53568 
4.52941 
4-52316 
4-51693 

.23203 
.23231 
-23260 
•23288 
•23316 

•97271 
•97264 
-97257 
.97251 
•97244 

.23854 
-23885 
-23916 
.23946 
-23977 

4-19215 
4.18675 
4.18137 
4.17600 
4-17064 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

•21644 
•21672 
•21701 
.21729 
.21758 

•97630 
•97623 
•97617 
.97611 
.97604 

.22169 
-22200 
-22231 
•22261 
•22292 

4-51071 
4-50451 
4.49832 
4-49215 
4-48600 

•23345 
-23373 
•23401 
•23429 
•23458 

•97237 
•97230 
.97223 
.97217 
•97210 

.  24008 
•24039 
.24069 
.24100 
.24131 

4.16530 
4-15997 
4-15465 
4-14934 
4-14405 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

•21786 
.21814 
•21843 
•21871 
•21899 

.97598 
.97592 
.97585 
.97579 
-97573 

.22322 
•22353 
-22383 
•22414 
.  22444 

4-47986 
4.47374 
4-46764 
4-46155 
4-45548 

23486 
•23514 
•23542 
•23571 
•23599 

•97203 
.97196 
.97189 
•97182 
-97176 

•24162 
•24193 
•24223 
•24254 
.24285 

4-13877 
4-13350 
4-12825 
4-12301 
4-11778 

25 
24 
23 
22 
21 

40 

41 
42 
48 
44 

.21928 
.21956 
•21985 
•22013 
.  22041 

•97566 
.97560 
.97553 
.97547 
•97541 

.22475 
•22505 
•22536 
•22567 
•22597 

4-44942 
4-44338 
4-43735 
4-43134 
4-42534 

•23627 
.23656 
•23684 
•23712 
.23740 

-97169 
•97162 
-97155 
-97148 
-97141 

.24316 
•24347 
•24377 
.  24408 
.  24439 

4-11256 
4-10736 
4-10216 
4-09699 
4-09182 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

•22070 
•22098 
•22126 
•22155 
.22183 

•97534 
.97528 
.97521 
.97515 
.97508 

.22628 
•22658 
•22689 
.22719 
-22750 

4.41936 
4-41340 
4.40745 
4-40152 
4-39560 

.23769 
•23797 
•23825 
•23853 
.23882 

-97134 
-97127 
-97120 
.97113 
-97106 

.  24470 
.24501 
.24532 
•24562 
•24593 

4.08666 
4-08152 
4-07639 
4-07127 
4-06616 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

•22212 
•22240 
•22268 
.22297 
•22325 

.97502 
.97496 
.97489 
•97483 
•97476 

•22781 
.22811 
.22842 
•22872 
•22903 

4-38969 
4-38381 
4.37793 
4-37207 
4-36623 

.23910 
•23938 
•23966 
.23995 
.24023 

.97100 
.97093 
-97086 
.97079 
-97072 

.24624 
.24655 
-24686 
.24717 
.24747 

4-06107 
4-05599 
4-05092 
4-04586 
4-04081 

10 

9 
8 

7 
6 

55 
56 
57 
58 
59 

.22353 
.22382 
•22410 
.22438 
.22467 

•97470 
-97463 
•97457 
•97450 
-97444 

.22934 
.22964 
•22995 
.23026 
•23056 

4-36040 
4-35459 
4-34879 
4-34300 
4-33723 

•24051 
•24079 
•24108 
•24136 
.24164 

-97065 
-97058 
-97051 
-97044 
-97037 

.24778 
-24809 
-24840 
•24871 
-  24902 

4-03578 
4.03076 
4.02574 
4-02074 
4-01576 

5 
4 
3 
2 
1 

60 

•22495 

•97437 

.23087 

4-33148 

.24192 

-97030 

-24933 

4-01078 

0 

' 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

/ 

77° 


212 


76° 


-NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
14°  15° 


t 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

o 

1 

2 
3 
4 

.24192 
.24220 
.  24249 
-24277 
.24305 

•97030 
•97023 
•97015 
-97008 
.97001 

•24933 
•24964 
.24995 
•25026 
•25056 

4-01078 
4-00582 
4-00086 
3-99592 
3-99099 

-25882 
-25910 
-25938 
.25966 
-25994 

•96593 
.96585 
-96578 
-96570 
-96562 

-26795 
-26826 
-26857 
-26888 
-26920 

3-73205 
3-72771 
3-72338 
3-71907 
3-71476 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.24333 
.24362 
.24390 
.24418 
.  24446 

-96994 
.96987 
-96980 
-96973 
.96966 

•25087 
.25118 
.25149 
.25180 
.25211 

3-98607 
3-98117 
3.97627 
3-97139 
3-96651 

-26022 
.26050 
-26079 
-26107 
-26135 

-96555 
-96547 
•96540 
•96532 
-96524 

-26951 
-26982 
-27013 
-27044 
-27076 

3-71046 
3-70616 
3-70188 
3-69761 
3-69335 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.  24474 
.24503 
.24531 
.24559 
.24587 

.96959 
.96952 
.96945 
-96937 
-96930 

.25242 
•25273 
.25304 
•25335 
.25366 

3-96165 
3-95680 
3-95196 
3-94713 
3-94232 

-26163 
-26191 
-26219 
-26247 
-26275 

-965,17 
•96509 
.96502 
.96494 
.96486 

-27107 
-27138 
-27169 
-27201 
•27232 

3-68909 
3-68485 
3-68061 
3-67638 
3-67217 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

•24615 
•24644 
.24672 
.24700 
.24728 

.96923 
-96916 
-96909 
-96902 
.96894 

•25397 
.25428 
•25459 
•25490 
.25521 

3-93751 
3-93271 
3-92793 
3-92316 
3-91839 

-26303 
-26331 
.26359 
-26387 

-26415 

-96479 
-96471 
-96463 
-96456 
-96448 

-27263 
-27294 
-27326 
-27357 
.27388 

3-66796 
3-66376 
3-65957 
3-65538 

3-65121 

45 
44 
43 
42 
-ii 

20 

21 
22 
23 
24 

.24756 
.24784 
•24813 
.  24841 
.24869 

•96887 
•96880 
-96873 
.96866 
-96858 

•25552 
•25583 
•25614 
•25645 
•25676 

3-91384 
3.90890 
3-90417 
3-89945 
3-89474 

-26443 
-26471 
-26500 
-26528 
-26556 

-96440 
-96433 
.96425 
-96417 
.96410 

.27419 
•27451 
.27482 
-27513 
-27545 

3-64705 
3-64289 
3-63874 
3-63461 
3  •  63048 

4O 

39 
38 
37 
36 

25 
26 
27 
28 
29 

24897 
24925 
24954 
24982 
25010 

-96851 
-96844 
-96837 
-96829 
-96822 

•25707 
•25738 
•25769 
•25800 
.25831 

3-89004 
3-88536 
3-88068 
3-87601 
3-87136 

-26584 
-26612 
-26640 
-26668 
-266P8 

-96402 
-96394 
-96386 
-96379 
-96371 

-27576 
-27607 
-27638 
-27670 
-27701 

3-62636 
3-62224 
3-61814 
3-61405 
3.60996 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

25038 
25066 
25094 
25122 
25151 

-96815 
-96807 
-96800 
-96793 
-96786 

•25862 
•25893 
•25924 
•25955 
•25983 

3-86671 
3-86208 
3-85745 
3-85284 
3-84824 

.26724 
26752 
26780 
26808 
26836 

-96363 
•96355 
-96347 
.96340 
-96332 

-27732 
.27764 
-27795 
.27826 
-27858 

3-60588 
3-60181 
3-59775 
3-59370 
3-58966 

30 

29 
28 
27 
28 

35 
36 
37 
38 
39 

25179 
25207 
25235 
25263 
25291 

•96778 
•96771 
-96764 
.96756 
•96749 

•26017 
•26048 
•26079 
.26110 

-26141 

3-84364 
3-83908 
3-83449 
3-82992 
3-82537 

26864 
26892 
26920 
26948 

26976 

-96324 
-96316 
•96308 
-96301 

-96293 

-27889 

•27921 
-27952 
.27983 
-28015 

3-58562 
3-58160 
3-57758 
3-57357 

3-56957 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

25320 
25348 
25376 
25404 
25432 

•96742 
•96734 
•96727 
•96719 
•96712 

•26172 
•26203 
•26235 
•26268 
•28297 

3-82083 
3-81630 
3-81177 
3-80726 
3-80278 

27004 
27032 
27060 
27088 
-27116 

-96285 
•96277 
•96269 
•96261 
•96253 

-28046 
•28077 
-28109 
-28140 
-28172 

3-56557 
3-56159 
3-55761 
3-55364 
3-54968 

20 

19 
18 
17 
18 

45 
46 
47 
48 
49 

25460 
25488 
25516 
25545 
25573 

•96705 
.96697 
•96690 
•96682 
•96675 

•26328 
.26359 
.26390 
.26421 
.26452 

3-79827 
3.79378 
3-78931 
3-78485 
3  -  78040 

-27144 
.27172 
-27200 
-27228 
-27256 

.96246 
•96238 
•96230 
-96222 
-96214 

-28203 
•28234 
-28266 
-28297 
-28329 

3-54573 
3-54179 
3-53785 
3-53393 
3-53001 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

25601 
25829 
25657 
25685 
25713 

•96667 
•96660 
•96653 
•96645 
•96638 

.26483 
•26515 
•26546 
•26577 
•26608 

3-77595 
3.77152 
3.76709 
3-76268 
3-75828 

-27284 
-27312 
-27340 
27368 
-27396 

-96206 
-96198 
-96190 
.96182 
96174 

-28360 
.28391 
.28423 
.28454 
-28486 

3-52609 
3-52219 
3-518?9 
3-51441 
3-51053 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

25741 
25769 
25798 
25826 
25854 

•96630 
.96623 
.96615 
•96608 
•96600 

.26639 
.26670 
•26701 
•26733 
•26764 

3.75388 
3.74950 
3.74512 
3.74075 
3-73640 

.  27424 
27452 
27480 
27508 
27536 

96166 
96158 
96150 
96142 
96134 

-28517 
.28549 
.28580 
•28612 
.28643 

3-50666 
3-50279 
3-49894 
3-49509 
3-49125 

5 
4 
3 
2 
1 

GO 

25882 

•96593 

.26795 

3-73205 

27564 

96126 

-28675 

3-48741 

0 

' 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

75° 


213 


74° 


TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
16°  17° 


• 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

' 

o 

1 

2 
3 

•  27564 
.27592 
.27620 
.27648 
.27676 

.96126 
.96118 
.96110 
.96102 
.96094 

.28675 
.28706 
.28738 
.28769 
.28800 

3-48741 
3-48359 
3-47977 
3-47596 
3-47216 

.29237 
-29265 
.29293 
.29321 
.29348 

.95630 
-95622 
-95613 
-95605 
.95596 

•30573 
-30605 
•30637 
.30669 
.30700 

3-27085 
3-26745 
3-26406 
3-26067 
3-25729 

60 

59 
58 
57 
56 

5 
6 
7 
8 

9 

.27704 
.27731 
.27759 
.27787 
.27815 

.96086 
.96078 
.96070 
.96062 
.96054 

.28832 
.28864 
.28895 
.28927 
.28958 

3-46837 
3-46458 
3-46080 
3-45703 
3.45327 

•29376 
.29404 
.29432 
.29460 
.29487 

.95588 
-95579 
-95571 
-95562 
-95554 

•30732 
.30764 
-30796 
.30828 
.30860 

3-25392 
3-25055 
3-24719 
3-24383 
3  •  24049 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.27843 
.27871 
.27899 
.27927 
.27955 

.96046 
•96037 
.96029 
.96021 
-96013 

.28990 
.29021 
.29053 
.29084 
.29116 

3-44951 
3-44576 
3-44202 
3-43829 
3-43456 

.29515 
.29543 
.29571 
.29599 
•29626 

.95545 
•95536 
-95528 
-95519 
-95511 

.30891 
.30923 
.30955 
.30987 
-31019 

3-23714 
3-23381 
3-23048 
3-22715 
3-22384 

5O 

49 
48 
47 
46 

15 
16 
17 
18 
19 

.27983 
.28011 
.28039 
.28067 
-28095 

.96005 
.95997 
.95989 
.95981 
.95972 

•29147 
•29179 
•29210 
.29242 
.29274 

3-43084 
3-42713 
3.42343 
3-41973 
3-41604 

.29654 
.29682 
.29710 
.29737 
.29765 

•95502 
•95493 
•95485 
-95476 
-95467 

.31051 
.31083 
-31115 
-31147 
.31178 

3-22053 
3  21722 
3-21392 
3-21063 
3-20734 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.28123 
.28150 
-28178 
.28206 
.28234 

.95964 
.95956 
.95948 
.95940 
.95931 

.29305 
.29337 
.29368 
.29400 
.29432 

3-41236 
3.40869 
3-40502 
3-40136 
3-39771 

.29793 
.29821 
.29849 
.29876 
.29904 

-95459 
-95450 
-95441 
•95433 
•95424 

.31210 

-31242 
-31274 
-31306 
.31338 

3-20406 
3-20079 
3-19752 
3-19426 
3-19100 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.28262 
-28290 
.28318 
.28346 
.28374 

.95923 
.95915 
.95907 
.95898 
.95890 

.29463 
•29495 
.29526 
.29558 
.29590 

3.39406 
3-39042 
3-38679 
3-38317 
3-37955 

.29932 
.29960 
.29987 
.30015 
.30043 

.95415 
.95407 
-95398 
-95389 
-95380 

.31370 
-31402 
.31434 
.31466 
.31498 

3-18775 
3-18451 
3-18127 
3-17804 
3-1748] 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.28402 
.28429 
.28457 
.28485 
.28513 

.95882 
.95874 
.95865 
.95857 
.95849 

.29621 
.29653 
.29685 
.29716 
.29748 

3.37594 
3-37234 
3-36875 
3-36516 
3-36158 

.30071 
-30098 
-30126 
-30154 
.30182 

.95372 
-95363 
•95354 
.95345 
.95337 

.31530 
.31562 
-31594 
.31626 
-31658 

3-17159 
3-16838 
3-16517 
3-16197 
3-15877 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.28541 
28569 
-28597 
.28625 
.28652 

.95841 
95832 
.95824 
.95816 
.95807 

.29780 
.29811 
.29843 
.29875 
.29906 

3-35800 
3-35443 
3-35087 
3-34732 
3-34377 

.30209 
-30237 
-30265 
.30292 
.30320 

.95328 
.95319 
.95310 
.95301 

-95293 

-31690 
-31722 
•31754 
•31786 
•31818 

3-15558 
3-15240 
3-14922 
3-14605 
3-14288 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

.28680 
28708 
.28736 
.28764 
.28792 

.95799 
.95791 
.95782 
•95774 
.95766 

.29938 
.29970 
.30001 
.30033 
.30065 

3-34023 
3-33670 
3-33317 
3-32965 
3-32614 

.30348 
•30376 
•30403 
-30431 
-30459 

.95284 
•95275 
•95266 
•95257 
.95248 

•31850 
.31882 
.31914 
.31946 
.31978 

3-13972 
3-13656 
3-13341 
3-13027 
3-12713 

2O 

19 
18 
17 
18 

45 
46 
47 
48 
49 

.28820 
28847 
28875 
28903 
28931 

•95757 
•95749 
.95740 
•95732 
.95724 

.30097 
.30128 
•30160 
.30192 
•30224 

3-32264 
3-31914 
3-31565 
3-31216 
3-30868 

•30486 
•30514 
-30542 
-30570 
.30597 

•95240 
•95231 
•95222 
-95213 
•95204 

.32010 
.32042 
-32074 
•32106 
-32139 

3-12400 
3-12087 
3-11775 
3-11464 
3-11153 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

28959 
28987 
29015 
29042 
29070 

.95715 
.95707 
•95698 
.95690 
.95681 

•30255 
•30287 
30319 
30351 
30382 

3-30521 
3-30174 
3-29829 
3-29483 
3-29139 

•30625 
.30653 
.30680 
.30708 
-30736 

•95195 
-95186 
.95177 
.95168 
•95159 

-32171 
.32203 
•32235 
.32267 
32299 

3-10842 
3-10532 
3-10223 
3-09914 
3-09606 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

29098 
29126 
29154 
29182 
29209 

.95673 
•95664 
.95656 
•95647 
.95639 

30414 
30446 
30478 
30509 
30541 

3-28795 
3-28452 
3-28109 
3-27767 
3-27426 

.30763 
30791 
30819 
30846 
30874 

.95150 
•95142 
.95133 
•95124 
•95115 

32331 
32363 
32396 
32428 
32460 

3-09298 
3-08991 
3-08685 
3-08379 
3-08073 

5 

3 
2 
1 

£0_ 

29237 

-95630 

30573 

3-27085 

30902 

.95106 

32492 

3-07768 

O 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

* 

73° 


214 


73° 


-NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
18°  19° 


' 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

o 
1 

2 
3 

4 

.30902 
.30929 
.30957 
.30985 
.31012 

.95106 
.95097 
.95088 
.95079 
.95070 

•32492 
.32524 
•32556 
.32588 
.32621 

3-07768 
3-07464 
3-07160 
3-06857 
3-06554 

-32557 
•32584 
-32612 
•32639 
-32667 

•94552 
•94542 
•94533 
•94523 
•94514 

•34433 
-34465 
-34498 
-34530 
.34563 

2-90421 
2-90147 
2-89873 
2-89600 
2-89327 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.31040 
.31068 
.31095 
.31123 
.31151 

.95061 
.95052 
.95043 
.95033 
.95024 

.32653 
.32685 
.32717 
•32749 
.32782 

3-06252 
3-05950 
3.05649 
3-05349 
3-05049 

-32694 
-32722 
•32749 
•32777 
•32804 

-94504 
•94495 
-94485 
•94476 
•  94466 

.34596 
.34628 
.34661 
-34693 
•34726 

2-89055 
2-88783 
2-88511 
2-88240 
2-87970 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.31178 
.31206 
.31233 
.31261 
.31289 

.95015 
.95006 
.94997 
.94988 
.94979 

.32814 
.32846 
.32878 
.32911 
.32943 

3  -  04749 
3-04450 
3-04152 
3-03854 
3-03556 

-32832 
.32859 
-32887 
•32914 
.32942 

-94457 

•  94447 
•94438 
•94428 
.94418 

•34758 
-34791 
•34824 
•34856 
.34889 

2-87700 
2.87430 
2.87161 
2-86892 
2.86624 

5O 

49 
48 
47 
46 

15 
16 
17 
18 
19 

•31316 
.31344 
•31372 
.31399 
•31427_ 

.94970 
.94961 
.94952 
.94943 
.94933 

•32975 
.33007 
.33040 
.33072 
.33104 

3-03260 
3-02963 
3-02667 
3-02372 
3-02077 

•32969 
.32997 
-33024 
,33051 
.33079, 

.94409 
•94399 
•94390 
•94380 
.94370 

.34922 
.34954 
•34987 
.35020 
.35052 

2.86356 
2-86089 
2-85822 
2-85555 
2-85289 

45 
44 
43 
42 
41 

30 

21 
22 
23 
24 

.31454 
.31482 
.31510 
.31537 
.31565 

.94924 
.94915 
.94906 
.94897 
.94888 

.33136 
.33169 
.33201 
.33233 
.33266 

3-01783 
3-01489 
3-01196 
3-00903 
3-00611 

•33106 
•33134 
-33161 
.33189 
.33216 

-94361 
.94351 
.94342 
.94332 
.94322 

.35085 
.35118 
.35150 
.35183 
.35216 

2-85023 
2-84758 
2-84494 
2-84229 
2-83965 

4O 

39 
38 
37 
36 

25 
26 
27 
28 
29 

•31593 
.31620 
•31648 
•31675 
.31703 

.94878 
.94869 
•94860 
.94851 
.94842 

.33298 
.33330 
.33363 
.33395 
.33427 

3-00319 
3-00028 
2-99738 
2-99447 
2-99158 

•33244 
-33271 
-33298 
-33326 
•33353 

.94313 

.94303 
.94293 
.94284 
.94274 

.35248 
.35281 
.35314 
.35346 
.35379 

2-83702 
2-83439 
2-83176 
2-82914 
2-82653 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

•31730 
•31758 
•31786 
•31813 
.31841 

.94832 
.94823 
.94814 
.94805 
.94795 

•33460 
.33492 
•33524 
•33557 
.33589 

2-98868 
2-98580 
2-98292 
2-98004 
2-97717 

•33381 

•33408 
•33436 
•33463 
•33490 

-94264 
-94254 
.94245 
.94235 
•94225 

-35412 

•35445 
-35477 
-35510 
•35543 

2-82391 
2-82130 
2-81870 
2-81610 
2-81350 

30 

29 
28 
27 
36 

35 
36 
37 
38 

39 

.31868 
.31896 
.31923 
•31951 
.31979 

.94786 
.94777 
.94768 
.94758 
.94749 

•33621 
•33654 
•33686 
-33718 
•33751 

2-97430 
2-97144 
2-96858 
2-96573 
2-9fi288 

-33518 
•33545 
•33573 
•33600 
.33627 

-94215 
-94206 
-94196 
-94186 

.94176 

-35576 
-35608 
-35641 
-35674 
-35707 

2-81091 
2-80833 
2-80574 
2-80316 

2-80059 

25 
24 
23 
22 
-21 

4O 

41 
42 
43 
44 

.32006 
.32034 
.32061 
.32089 
.32116 

.94740 
•94730 
•94721 
•94712 
.94702 

•33783 
•33818 
.33848 
•33881 
.33913 

2-96004 
2-95721 
2-95437 
2-95155 
2-94872 

-33655 
•33682 
•33710 
•33737 
•33764 

.94167 
-94157 
-94147 
-94137 
-94127 

-35740 
-35772 
•35805 
•35838 
.35871 

2-79802 
2-79545 
2-79289 
2-79033 
2-78778 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

.32144 
.32171 
.32199 
.32227 
•32254 

.94693 
.94684 
•94674 
•94665 
.94656 

•33945 
•33978 
•34010 
.  34043 
.  34075 

2-94591 
2-94309 
2-94028 
2-93748 
2-93468 

-33792 
•33819 
•33846 
•33874 
•33901 

.94118 
.94108 
-94098 
-94088 
-94078 

.35904 
.35937 
.35969 
.36002 
.36035 

2-78523 
2-78269 
2-78014 
2-77761 
2-77507 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

•32282 
.32309 
.32337 
.32364 
.32392 

.94646 
.94637 
•94627 
•94618 
.94609 

•34108 
•34140 
•34173 
•34205 
-34238 

2-93189 
2.92910 
2.92632 
2.92354 
2-92076 

-33929 
•33956 
•33983 
-34011 
-34038 

-94068 
-94058 
•  94049 
•94039 
•94029 

-36068 
.36101 
-36134 
•36167 
-36199 

2-77254 
2-77002 
2-76750 
2-76498 
2-76247 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•32419 
•32447 
•32474 
•32502 
.32529 

.94599 
.94590 
.94580 
.94571 
.94561 

•34270 
•34303 
-34335 
•34368 
.  34400 

2.91799 
2-91523 
2-91246 
2-90971 
2-90696 

-34065 
•34093 
-34120 
•34147 
•34175 

-94019 
-94009 
.93999 
.93989 
•93979 

-36232 
•36265 
.36298 
.36331 
•36364 

2-75996 
2-75746 
2-75496 
2.75246 
2-74997 

5 
4 
3 

2 

80 

.32557 

.94552 

.34433 

2-90421 

.34202 

-93969 

.36397 

2  -  74748 

O 

Cos. 

Sin. 

Cot. 

Tan. 

COP. 

Sin. 

Cot. 

Tan. 

71° 


215 


70° 


TABLE  10  —NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
30°  31° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

t 

60 

59 
58 
57 
56 
55 
54 
53 
52 
51 

o 

1 

2 
3 
4 

-34202 
.34229 
.34257 
.34284 
•34311 

•93969 
•93959 
•93949 
•93939 
.93929 

•36397 
•36430 
•36463 
•36496 
.36529 

2-74748 
2.74499 
2-74251 
2-74004 
2-73756 

•35837 
•35864 
.35891 
.35918 
.35945 

-93358 
-93348 
-93337 
.93327 
.93316 

-38386 
-38420 
-38453 
-38487 
•38520 

2-60509 
2-60283 
2-60057 
2-59831 
2-59606 

5 
6 
7 
8 
9 
10 

12 
13 
14 

15 
16 
17 
18 
19_ 
30 
21 
22 
23 
24 

.34339 
.34366 
.34393 
.34421 
.34448 

•93919 
.93909 
.93899 
.93889 
.93879 

•36562 
•36595 
.36628 
•36661 
.36694 

2-73509 
2-73263 
2-73017 
2-72771 
2-72526 

•35973 
-36000 
-36027 
-36054 
-36081 

•93306 
.93295 
.93285 
•93274 
-93264 

-38553 
-385S7 
-38620 
-38654 
•38687 

2-59381 
2-59156 
2-58932 
2.58708 
2-58484 

.34475 
.34503 
.34530 
.34557 
-34584 

.93869 
.93859 
.93849 
.93839 
.93829 

•36727 
.36760 
.36793 
.36826 
.36859 

2-72281 
2-72036 
2-71792 
2-71548 
2.71305 

•36108 
•36135 
.36162 
•36190 
•36217 

-93253 
•93243 
.93232 
-93222 
-93211 

•38721 
•38754 
-38787 
-38821 
.38854 

2-58261 
2-58038 
2-57815 
2.57593 
2-57371 

50 

49 
48 
47 
46 

.34612 
34639 
-34666 
.34694 
.34721 

.93819 
.93809 
•93799 
.93789 
.93779 

.36892 
.36925 
.36958 
•36991 

.37024 

2-71062 
2-70819 
2-70577 
2-70335 
2-70094 

.36244 
•36271 
-36298 
.36325 
.36352 

.93201 
.93190 
•93180 
.93169 
-93159 

-38888 
-38921 
-38955 
•38988 

.39022 

2-57150 
2.56928 
2-56707 
2-56487 

2-56266 

45 
44 
43 
42 
41 

•34748 
.34775 
.34803 
.34830 
.34857 

.93769 
.93759 
.93748 
.93738 
.93728 

-37057 
•37090 
•37123 
.37157 
.37190 

2-69853 
2-69612 
2-69371 
2-69131 
2-68892 

.36379 
-36406 
-36434 
.36461 
-36488 

-93148 
-93137 
-93127 
-93116 
-93106 

•39055 
-39089 
-39122 
-39156 
-39190 

2-56046 
2-55827 
2-55608 
2-55389 
2-55170 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 

•34884 
•34912 
.34939 
.34966 
.34993 

.93718 
.93708 
.93698 
.93688 
.93677 

.37223 
.37256 
.37289 
.37322 
.37355 

2-68653 
2-68414 
2-68175 
2-67937 
2-67700 

-36515 
-36542 
.36569 
.36596 
.36623 

.93095 
•93084 
.93074 
.93063 
.93052 

-39223 
39257 
.39290 
-39324 
-39357 

2-54952 
2.54734 
2-54516 
2.54299 
2-54082 

35 
34 
33 
32 
31 
30 
29 
28 
27 
26 

.35021 
.35048 
.35075 
.35102 
-35130 

.93667 
.93657 
•93647 
•93637 
.93626 

.37383 

.37422 
-37455 
•37488 
-37521 

2.67462 
2.67225 
2.66989 
2.66752 
2-66516 

.36650 
.36677 
-36704 
-36731 
-36758 

.93042 
.93031 
•93020 
•93010 
-92999 

39391 
-39425 
•39458 
-39492 
-39526 

2-53865 
2-53648 
2-53432 
2-53217 
2-53001 

•35157 
•35184 
•35211 
•35239 

.35266 

.93616 
•93606 
•93596 
•93585 
.93575 

•37554 
•37588 
•37621 
•37654 
.37687 

2-66281 
2-66046 
2.65811 
2.65576 
2-65342 

-36785 
-36812 
-36839 
-36867 
-36894 

.92988 
-92978 
-92967 
-92956 
-92945 

-39559 
-39593 
-39626 
-39660 
-39694 

2-52786 
2-52571 
2-52357 
2.52142 
2-51929 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

•35293 
.35320 
•35347 
•35375 
•35402 

.93565 
.93555 
.93544 
.93534 
.93524 

.37720 
•37754 
-37787 
-37820 
-37853 

2-65109 
2-64875 
2.64642 
2-64410 
2-64177 

-36921 
-36948 
-36975 
-37002 
-37029 

-92935 
-92924 
-92913 
•92902 
.92892 

-39727 
-39761 
•39795 
•39829 
.39862 

2.51715 
2.51502 
2-51289 
2.51076 
2-50864 

30 

19 
18 
17 
16 
15 
14 
13 
12 
11 

1O 

9 
8 
7 
6 
5 
4 
3 
2 
1 

-£ 

45 
46 
47 
48 
49 

•35429 
•35456 
.35484 
•35511 
35538 

•93514 
.93503 
•93493 
•93483 
.93472 

.37887 
.37920 
.37953 
•37986 
-38020 

2.63945 
2-63714 
2.63483 
2-63252 
2-63021 

-37056 
-37083 
.37110 
.37137 
.37164 

-92881 
-928VO 
-92859 
-92849 
-92838 

-39896 
•39930 
-39963 
•39997 
.40031 

2-50652 
2  -  50440 
2-50229 
2-50018 
2-49807 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60. 

35565 
35592 
35619 
35647 
35674 

•93462 
•93452 
•93441 
-93431 
.93420 

.38053 
•38086 
.38120 
.38153 
38186 

2.62791 
2-62561 
2-62332 
2-62103 
2-61874 

.37191 
.37218 
.37245 
.37272 
-37299 

•92827 
.92816 
.92805 
.92794 
.92784 

-40065 
.40098 
-40132 
-40166 
-  40200 

2-49597 
2-49386 
2-49177 
2-48967 
2-48758 

35701 
35728 
35755 
35782 
35810 

•93410 
.93400 
•93389 
.93379 
.93368 

•38220 
38253 
38286 
38320 
-38353 

2-61646 
2.61418 
2-61190 
2-60963 
2-60736 

-37326 
-37353 
-37380 
37407 
37434 

.92773 
-92762 
-92751 
-92740 
92729 

-40234 
40267 
40301 
40335 
40369 

2-48549 
2-48340 
2-48132 
2-47924 
2-47716 

35837 

•93358 

-38386 

2-60509 

37461 

92718 

40403 

2-47509 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

216 


68° 


TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS, 
22°  23° 


1 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

, 

0 

2 
8 
4 

.37461 
.37488 
•37515 
.37542 
.37569 

.92718 
.92707 
•92697 
.92686 
.92675 

.40403 
.40436 
.40470 
.40504 
.40538 

2-47509 
2.47302 
2.47095 
2-46888 
2-46682 

.39073 
.39100 
-39127 
-39153 
.39180 

•92050 
.92039 
.92028 
-92016 
-92005 

•42447 
.42482 
.42516 
.42551 
.42585 

2-35585 
2.35395 
2-35205 
2-35015 
2-34825 

60 

59 
58 
57 
5ft 

5 
6 
7 
8 
9 

•37595 
.37622 
•37649 
.37676 
.37703 

.92664 
.92653 
.92642 
.92631 
92620 

.40572 
.40606 
.40640 
.40674 
.40707 

2-46476 
2-46270 
2-46065 
2.45860 
2.45655 

•39207 
-39234 
.39260 
•39287 
.39314 

•91994 
.91982 
.91971 
.91959 
•91948 

.42619 
.42654 
•42688 
-42722 
•42757 

2-34636 
2-34447 
2-34258 
2-34069 
2-33881 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

•37730 
.37757 
•37784 
.37811 
.37838 

.92609 
.92598 
.92587 
.92576 
.92565 

.40741 
.40775 
.40809 
.40843 
.40877 

2-45451 
2-45246 
2-45043 
2.44839 
2.44636 

•39341 
.39367 
•39394 
-39421 
.39448 

.91936 
-91925 
.91914 
.91902 
.91891 

-42791 
•42826 
.42860 
•42894 
.42929 

2-33693 
2-33505 
2-33317 
2-33130 
2-32943 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

.37865 
.37892 
.37919 
.37946 
.37973 

.92554 
.92543 
.92532 
.92521 
.92510 

.40911 
.40945 
.40979 
.41013 
.41047 

2-44433 
2-44230 
2-44027 
2-43825 
2-43623 

.39474 
.39501 
.39528 
.39555 
.39581 

.91879 
.91868 
.91856 
.91845 
.91833 

•42963 
.42998 
.43032 
.43067 
.43101 

2-32756 
2-32570 
2-32383 
2-32197 
2.32012 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.37999 
.38026 
.38053 
.38080 
.38107 

.92499 
.92488 
.92477 
.92466 
.92455 

.41081 
.41115 
.41149 
.41183 
.41217 

2-43422 
2-43220 
2-43019 
2-42819 
2-42618 

•39608 
•39635 
•39661 
.39688 
.39715 

-91822 
.91810 
.91799 
.91787 
.91775 

.43136 
.43170 
.43205 
-43239 
-43274 

2-31826 
2.31641 
2-31456 
2-31271 
2-31086 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.38134 
.38161 
.38188 
-38215 
.38241 

.92444 
.92432 
.92421 
.92410 
.92399 

.41251 
.41285 
.41319 
.41353 
.41387 

2-42418 
2-42218 
2-42019 
2.41819 
2-41620 

-39741 
.39768 
.39795 
.39822 
.39848 

.91764 
.91752 
.91741 
.91729 
.91718 

-43308 
.43343 
•43378 
-43412 
-43447 

2-30902 
2-30718 
2-30534 
2-30351 
2-30167 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.38268 
.38295 
.38322 
.33349 
.38376 

.92388 
.92377 
•92366 
.92355 
.92343 

.41421 
.41455 
.41490 
.41524 
.41558 

2  41421 
2-41223 
2-41025 
2-40827 
2-40629 

•39875 
-39902 
.39928 
-39955 
.39982 

-91706 
.91694 
.91683 
.91671 
.91660 

-43481 

.43516 
.43550 
-43585 
.43620 

2.29984 
2-29801 
2.29619 
2-29437 
2-29254 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

-38403 
.38430 
.38456 
.38483 
.38510 

.92332 
.92321 
.92310 
.92299 
.92287 

-41592 
.41626 
.41660 
.41694 
.41728 

2-40432 
2-40235 
2-40038 
2-39841 

2-39645 

.40008 
•40035 
•40062 
•40088 
.40115 

.91648 
.91636 
.91625 
.91613 
.91601 

.43654 
.43689 
.43724 
.43758 
.43793 

2-29073 
2-28891 
2-28710 
2-28528 
2.28348 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

.38537 

.38564 
.38591 
•38617 
.38644 

.92276 
.92265 
.92254 
.92243 
.92231 

.41763 
.41797 
.41831 
.41865 
.41899 

2-39449 
2.39253 
2-39058 
2-38863 
2.38668 

.40141 
•40168 
•40195 
-40221 
.40248 

.91590 
.91578 
.91566 
.91555 
.91543 

.43828 

.43862 
•43897 
-43932 
.43966 

2-28167 
2-27987 
2.27806 
2-27626 
2-27447 

2O 

19 
18 
17 
16 

45 
46 
47 
48 
49 

-38671 
.38698 
.38725 
.38752 
-38778 

.92220 
.92209 
.92198 
.92186 
.92175 

.41933 
.41968 
.42002 
.42036 
.42070 

2.38473 
2-38279 
2-38084 
2-37891 
2-37697 

-40275 
40301 
-40328 
.40355 
.40381 

.91531 
.91519 
.91508 
.91496 
.91484 

-44001 
.44036 
-44071 
.44105 
.44140 

2-27267 
2-27088 
2-26909 
2-26730 
2-26552 

15 
14 
13 
12 
11 

5O 

51 
52 
53 
54 

.38805 
.38832 
.38859 
.38886 
.38912 

.92164 
.92152 
.92141 
.92130 
.92119 

.42105 
.42139 
.42173 
.42207 
.42242 

2-37504 
2-37311 
2-37118 
2-36925 
2.36733 

.40408 
•40434 
.40461 
.40488 
•40514 

.91472 
.91461 
.91449 
.91437 
.91425 

-44175 
.44210 
.44244 
.44279 
.44314 

2-26374 
2-26196 
2-26018 
2-25840 
2-25663 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•38939 
•38966 
•38993 
.39020 
.39046 

.92107 
.92096 
•92085 
.92073 
.92062 

•42276 
.42310 
.42345 
.42379 
.42413 

2-36541 
2.36349 
2.36158 
2-35967 
2-35776 

.40541 
•40567 
.40594 
.40621 
.40347 

.91414 
.91402 
.91390 
.91378 
.91366 

.44349 
.44384 
.44418 
.44453 
.44488 

2-25486 
2-25309 
2-25132 
2-24956 
2-24780 

5 
"4 
3 

2 

60 

.39073 

.92050 

•42447 

2-35585 

.40674 

.91355 

.44523 

2-24604 

O 

1 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

67° 


217 


66° 


-NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
34°  25° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

f 

0 

1 

2 
3 
4 

.40674 
.40700 
.40727 
.40753 
.40780 

.91355 
.91343 
.91331 
.91319 
.91307 

•44523 
•44558 
•44593 
•44627 
.44662 

2.24604 
2-24428 
2.24252 
2.24077 
2.23902 

-42262 
.42288 
.42315 
.42341 
-42367 

•90631 
-90618 
•90606 
•90594 
.90582 

.46631 
•46666 
•46702 
•46737 
•46772 

2-14451 
2-14288 
2.14125 
2-13963 
2-13801 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.40806 
.40833 
.40860 
-40886 
.40913 

.91295 
•91283 
.91272 
.91260 
.91248 

•44697 
•44732 
•44767 
.44802 
•44837 

2-23727 
2.23553 
2.23378 
2-23204 
2-23030 

.42394 
•42420 
.42446 
•42473 
-42499 

-90569 
•90557 
.90545 
-90532 
.90520 

.46808 
-46843 
.46879 
•46914 
.46950 

2-13639 
2.13477 
2-13316 
2.13154 
2-12993 

55 

54 
53 
52 
51 

10 

11 
12 
13 
14 

.40939 
.40966 
•40992 
.41019 
.41045 

.91236 
.91224 
.91212 
.91200 
.91188 

•44872 
.44907 
.44942 
.44977 
.45012 

2-22857 
2.22683 
2-22510 
2.22337 
2.22164 

•42525 
.42552 
.42578 
.42604 
.42631 

.90507 
.90495 
-90483 
•90470 
.90458 

.46985 
.47021 
.47056 
.47092 
-47128 

2-12832 
2.12671 
2-12511 
2-12350 
2-12190 

50 

49 
48 

47 
46 

15 
16 
17 
18 
19 

•41072 
.41098 
.41125 
.41151 
.41178 

.91176 
.91164 
.91152 
.91140 
.91128 

.45047 
.45082 
.45117 
•45152 
.45187 

2.21992 
2.21819 
2.21647 
2.21475 
2-21304 

-42657 
.42683 
-42709 
-42736 
-42762 

.  90446 
•90433 
•90421 
.90408 
-90396 

.47163 
.47199 
.47234 
-47270 
.47305 

2-12030 
2-11871 
2-11711 
2-11552 
2-11392 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

•41204 
•41231 
.41257 
.41284 
•41310 

-91116 
•91104 
.91092 
.91080 
.91068 

.45222 
•45257 
.45292 
•45327 
.45362 

2-21132 
2.20961 
2-20790 
2-20619 
2-20449 

•42788 
-42815 
.42841 
-42867 
.42894 

.90383 
•90371 
.90358 
.90346 
.90334 

-47341 
-47377 
.47412 
.47448 
.47483 

2-11233 
2-11075 
2-10916 
2-10758 
2-10600 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.41337 
•41363 
•41390 
•41416 
•41443 

•91056 
•91044 
•91032 
•91020 
.91008 

•45397 

-45432 
.45467 
•45502 
.45538 

2-20278 
2.20108 
2-19938 
2-19769 
2-19599 

-42920 
-42946 
•42972 
-42999 
.43025 

-90321 
.90309 
.90296 
-90284 
.90271 

-47519 
-47555 
.47590 
.47626 
-47662 

2-10442 
2-10284 
2-10126 
2-09969 
2-09811 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

•41469 
.41496 
.41522 
.41549 
.41575 

.90996 
.90984 
.90972 
.90960 
.90948 

.45573 
•45608 
.45643 
-45678 
.45713 

2.19430 
2.19261 
2-19092 
2-18923 
2-18755 

.43051 
.43077 
•43104 
.43130 
-43156 

-90259 
.90246 
.90233 
.90221 
•90208 

.47698 
.^7733 
.47769 
.47805 
.47840 

2-09654 
2-09498 
2-09341 
2-09184 
2-09028 

30 

29 
28 
27 
26 

35 
36 
37 
38 

39 

•41602 
•41628 
.41655 
•41681 

•41707 

90936 
•90924 
-90911 
•90899 
-90887 

•45748 
•45784 
•45819 
•45854 
.45889 

2-18587 
2-18419 
2-18251 
2-18084 
2-17916 

•43182 

.43209 
•43235 
•43261 
-43287 

-90196 
-90183 
-90171 
•90158 
.90146 

.47876 
.47912 
.47948 
.47984 
.48019 

2-08872 
2-08716 
2-08560 
2-08405 
2-08250 

25 
24 
23 
22 
-21 

40 

41 
42 
43 
44 

•41734 
•41760 
•41787 
•41813 
•41840 

•90875 
.90863 
-90851 
-90839 
-90826 

•45924 
.45960 
•45995 
•46030 
.46065 

2-17749 
2-17582 
2-17416 
2-17249 
2.17083 

-43313 
.43340 
•43366 
•43392 
.43418 

.90133 
.90120 
.90108 
•90095 
-90082 

.48055 
-48091 
.48127 
.48163 
.48198 

2-08094 
2-07939 
2-07785 
2.07630 
2-07476 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

•41866 
.41892 
.41919 
•41945 
.41972 

•90814 
•90802 
.90790 
•90778 
.90766 

.46101 
•46136 
.46171 
•46206 
•46242 

2.16917 
2-16751 
2-16585 
2-16420 
2-16255 

•  43445 
•43471 
•43497 
•43523 
.43549 

•90070 
•90057 
.90045 
-90032 
-90019 

.48234 
.48270 
-48306 
-48342 
.48378 

2-07321 
2-07167 
2-07014 
2-06860 
2-06706 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

•41998 
.42024 
•42051 
.42077 
.42104 

•90753 
-90741 
.90729 
.90717 
•90704 

.46277 
•46312 
•46348 
•46383 
•46418 

2-16090 
2-15925 
2-15760 
2-15596 
2-15432 

•43575 
•43602 
•43628 
•43654 
.43680 

.90007 
•89994 
•89981 
-89968 
.89956 

.48414 
.48450 
.48486 
.48521 
.48557 

2-06553 
2.06400 
2-06247 
2-06094 
2-05942 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•42130 
•42156 
.42183 
.42209 
.42235 

•90692 
•90680 
•90668 
.90655 
.90643 

.46454 
.46489 
•46525 
•46560 
.46595 

2-15268 
2-  15104* 
2-14940 
2-14777 
2-14614 

•43706 
•43733 
-43759 
•43785 
.43811 

.89943 
.89930 
-89918 
.89905 
.89892 

.48593 
.48629 
.48665 
•48701 
•48737 

2-05790 
2.05637 
2-05485 
2-05333 
2-05182 

5 

4 
3 
2 
1 

60 

.42262 

.90631 

-46631 

2-14451 

•43837 

.89879 

.48773 

2-05030 

_o 

/ 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

/ 

65° 


218 


64° 


-NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
26°  27° 


* 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

t 

0 

2 
3 

4 

•43837 
.43863 
.43889 
.43916 
.43942 

.89879 
•89867 
.89854 
.89841 
.89828 

.48773 
.48809 
.48845 
.48881 
.48917 

2.05030 
2.04879 
2.04728 
2-04577 
2-04426 

.45399 

•45425 
.45451 
.45477 
.45503 

-89101 
-89087 
-89074 
-89061 
.89048 

-50953 
-50989 
-51026 
-51063 
-51099 

1-96261 
1-98120 
1-95979 
1.95838 
1-95698 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.43968 
.43994 
.44020 
.  44046 
.44072 

.89816 
.89803 
.89790 
.89777 
.89764 

.48953 
.48989 
.49026 
.49062 
.49098 

2-04276 
2.04125 
2.03975 
2.03825 
2-03675 

.45529 
.45554 
.45580 
.45606 
.45632 

-89035 
-89021 
.89008 
.88995 
.88981 

.51136 
.51173 
.51209 
.51246 
•51283 

1-95557 
1-95417 
1-95277 
1-95137 
1-94997 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.44098 
.44124 
.44151 
.44177 
.44203 

.89752 
.89739 
.89726 
.89713 
.89700 

.49134 
.49170 
.49206 
.49242 
.49278 

2.03526 
2.03376 
2-03227 
2-03078 
2-02929 

.45658 
-45684 
-45710 
-45736 
.45762 

•88968 
-88955 
-88942 
.88928 
-88915 

-51319 
-51356 
.51393 
-51430 
-51467 

1-94858 
1-94718 
1-94579 
1-94440 
1-94301 

5O 

49 
48 
47 
46 

15 
16 
17 
18 
19 

•44229 
•44255 
•44281 
.44307 
.44333 

.89687 

.89674 
.89662 
.89649 
.89636 

.49315 
.49351 
.49387 
•49423 
.49459 

2.02780 
2.02631 
2-02483 
2.02335 
2-02187 

.45787 
.45813 
.45839 
.45865 
.45891 

.88902 
.88888 
.88875 
-88862 
.88848 

-51503 
-51540 
-51577 
-51614 
.51651 

1.94162 
1.94023 
1.93885 
1.93746 
1.93608 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.44359 
.44385 
.44411 
.44437 
.44464 

.89623 
.89610 
.89597 
.89584 
.89571 

•49495 
.49532 
.49568 
.49604 
.49640 

2-02039 
2-01891 
2.01743 
2-01596 
2-01449 

-45917 
.45942 
.45968 
•45994 
.46020 

-88835 
•88822 
.88808 
•88795 
.88782 

•51688 
•51724 
•51761 
.51798 
.51835 

1.93470 
1-93332 
1.93195 
1.93057 
1-9?920 

40 

39 
38 
37 
36 

25 
2fi 
27 
28 
29 

•44490 
•44516 
.44542 
•44568 
.44594 

.89558 
.89545 
.89532 
.89519 
.89506 

•49677 
.49713 
.49749 
.49786 
-49822 

2.01302 
2-01155 
2.01008 
2.00862 
2-00715 

.46046 
.46072 
-46097 
-46123 
-46149 

-88768 
-88755 
.88741 
.88728 
.88715 

.51872 
-51909 
-51946 
-51983 
-52020 

1.92782 
1.92645 
1.92508 
1.92371 
1.92235 

35 
34 
33 
32 
31 

30 

81 
82 
33 
34 

.44620 
.44646 
.44672 
.44698 
44724 

•89493 
.89480 
.89467 
.89454 
.89441 

.49858 

.49894 
.49931 
.49967 
.  50004 

2.00569 
2-00423 
2-00277 
2-00131 
1.99986 

-46175 
-46201 
-46226 
.46252 
.46278 

.88701 
.88688 
.88674 
.88661 
.88647 

-52057 
-52094 
-52131 
.52168 
.52205 

1.92098 
1-91962 
1.91826 
1.91690 
1.91554 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

44750 
44776 
44802 
44828 
44854 

.89428 

.89415 
.89402 
.89389 
.89376 

.  50040 
.50076 
•50113 
.50149 
.50185 

1.99841 
1.99695 
1.99550 
1.99406 
1.99261 

.46304 
-46330 
-46355 
-46381 
.46407 

.88634 

•88620 
.88607 
.88593 
.88580 

.52242 
.52279 
.52316 
.52353 
.52390 

1.91418 
1-91282 
1.91147 
1.91012 

1-90876 

25 
24 
23 
22 
-21 

!i° 

42 
43 
44 

44880 
44906 
44932 
44958 
44984 

.89363 
.89350 
.89337 
.89324 
.89311 

.50222 
.50258 
.50295 
.50331 
.50368 

1-99116 
1.98972 
1.98828 
1.98684 
1.98540 

-46433 
•46458 
.46484 
.46510 
.46536 

•88566 
•88553 
•88539 
.88526 
.88512 

.52427 
.52464 
-52501 
-•52538 
-52575 

1-90741 
1-90607 
1-90472 
1-90637 
1-90203 

30 

19 
18 
17 
16 

45 
46 
47 
48 
49 

45010 
45036 
45062 
45088 
45114 

.89298 
.89285 
.89272 
.89259 
.89245 

.  50404 
.  50441 
•50477 
.50514 
.50550 

1-98396 
1.98253 
1.98110 
1-97966 
1-97823 

.46561 
.46587 
•46613 
46639 
.46664 

-88499 
.88485 
.88472 
.88458 
.88445 

-52613 
•52650 
52687 
52724 
.52761 

1-90069 
1-89935 
1-89801 
1-89667 
1.89533 

15 

14 
13 
12 
11 

50 

51 
52 
53 
54 

45140 
45166 
45192 
45218 
45243 

.89232 
.89219 
.89206 
.89193 
.89180 

.50587 
.50623 
50660 
.50696 
.50733 

1.97681 
1.97538 
1.97395 
1.97253 
1.97111 

-46690 
46716 
46742 
46767 
46793 

-88431 
.88417 
.88404 
.88390 
-88377 

52798 
52836 
52873 
52910 
52947 

1-89400 
1-99266 
1-89133 
1.89000 
1.88867 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

45269 
45295 
45321 
45347 
45373 

•89167 
.89153 
.89140 
•89127 
.89114 

•50769 
.50806 
.50843 
.50879 
50916 

1-96969 
1-96827 
1.96685 
1.96544 
1-96402 

46819 
46844 
46870 
46896 
46921 

88363 
88349 
88336 
88322 
88308 

52985 
53022 
53059 
53096 
53134 

1-88734 
1-88602 
1-88469 
1-88337 
1-88205 

5 
4 
3 

2 
1 

60 

45399 

.89101 

.50953 

1.96261 

46947 

88295 

53171 

1-88073 

_2 

"7~ 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

ft 

*° 

21 

9 

G 

J° 

-NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
28°  29°  i 


9 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

1 

3 

.46947 
.46973 
.46999 
.47024 
.47050 

.88295 
.88281 
.88267 
.88254 
.88240 

•53171 
.53208 
53246 
.53283 
.53320 

1-88073 
1-87941 
1-87809 
1-87677 
1.87546 

•48481 
.48506 
.48532 
.48557 
.48583 

87462 
.87448 
•87434 
87420 
.87406 

.55431 
•55469 
-55507 
-55545 
.55583 

i  80405 
1-80281 
1.80158 
1-80034 
1-79911 

60 

59 
58 
57 
56 

5 
6 

7 
8 
9 

.47076 
.47101 
•47127 
.47153 
.47178 

.88226 
88213 
.88199 
.88185 
.88172 

•53358 
.53395 
.53432 
53470 
.53507 

1-87415 
1-87283 
1-87152 
1-87021 
1.86891 

-48608 
.48634 
-48659 
.48684 
-48710 

-87391 
-87377 
-87363 
-87349 
.87335 

.55621 
•55659 
•55697 
•55736 
.55774 

1  79788 
1  79665 
1.79542 
1-79419 
1-79296 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.47204 
.47229 
.47255 
.47281 
.47306 

.88158 
.88144 
.88130 
.88117 
.88103 

.53545 
.53582 
.53620 
.53657 
.53694 

1.86760 
1.86630 
1.86499 
1.86369 
1.86239 

.48735 
•48761 
.48786 
-48811 
•48837 

.87321 
.87306 
.87292 
.87278 
.87264 

•55812 
•55850 
.55888 
.55926 
•55964 

1-79174 
1.79053 
1-78929 
1-78807 
1.78685 

50 

49 
48 
47 
46 

15 
16 
17 
18 
J9 

.47332 
•47358 
.47383 
.47409 
.47434 

.88089 
.88075 
.88062 
.88048 
.88034 

•53732 
.53769 
.53807 
.53844 
.53882 

1-86109 
1.85979 
1.85850 
1-85720 
1-85591 

.48862 
.48888 
.48913 
.48938 
-48964 

.87250 
.87235 
.87221 
.87207 
.87193 

.56003 
-56041 
-56079 
.56117 

-56156 

1-78563 
1  -  78441 
1-78319 
1-78198 
1.78077 

45 
44 
43 
42 
-41 

20 

21 
22 
23 
24 

.47460 
.47486 
.47511 
.47537 
.47562 

.88020 
.88006 
.87993 
.87979 
.87965 

.53920 
.53957 
.53995 
.54032 
.54070 

1-85462 
1-85333 
1-85204 
1-85075 
1-84946 

.48989 
.49014 
•49040 
•49065 
.49090 

-87178 
.87164 
.87150 
.87136 
.87121 

.56194 
-56232 
-56270 
.56309 
.56347 

1-77955 
1-77834 
1-77713 
1-77592 
1.77471 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.47588 
•47614 
•  47639 
.47665 
.47690 

.87951 
•87937 
•87923 
.87909 
.87896 

.54107 
.54145 
•54183 
.  54220 
.54258 

1.84818 
1-84689 
1-84561 
1-84433 
1-84305 

-49116 
.49141 
.49166 
-49192 
.49217 

-87107 
87093 
-87079 
-87064 
.87050 

-56385 

-56424 
-56462 
-56501 
.56539 

1-77351 
1-77230 
1.77110 
1-76990 
1.76869 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.47716 
.47741 
.47767 
.47793 
.47818 

.87882 
.87868 
.87854 
.87840 
.87826 

•54296 
.54333 
.54371 
.  54409 
.  54446 

1-84177 
1  .  84049 
1.83922 
1-83794 
1.83667 

.49242 
-49268 
.49293 
.49318 
.49344 

.87036 
.87021 
.87007 
.86993 
.86978 

.56577 
-56616 
.56654 
.56693 
-56731 

1  -  76749 
1-76629 
1-76510 
1-76390 
1-76271 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.47844 
.47869 
.47895 
.47920 
.47946 

.87812 
.87798 
.87784 
.87770 
.87756 

.  54484 
•54522 
.54560 
.54597 
.54635 

1.83540 
1-83413 
1-83286 
1-83159 
1.83033 

.49369 
.49394 
.49419 
.49445 
.49470 

-86964 
-86949 
-86935 
-86921 
-86906 

-56769 
-56808 
-56846 
-56885 

.56923 

1-76151 
1-76032 
1-75913 
1-75794 
1.75675 

25 
24 
23 
22 
-21 

40 

41 
42 
43 
44 

.47971 
.47997 
.48022 
.48048 
.48073 

•87743 
•87729 
.87715 
.87701 
.87687 

.  54673 
.54711 
.54748 
..54786 
.54824 

1-82906 
1-82780 
1.82654 
1.82528 
1-82402 

-49495 
-49521 
.49546 
-49571 
.49596 

.86892 
-86878 
.86863 
-86849 
.86834 

•56962 
.57000 
-57039 
-57078 
-57116 

1.75556 
1-75437 
1-75319 
1-75200 
1-75082 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

.48099 
.48124 
.48150 
.48175 
.48201 

.87673 
.87659 
.87645 
.87631 
.87617 

•54862 
.54900 
.54938 
.54975 
.55013 

1-82276 
1-82150 
1-82025 
1-81899 
1.81774 

-49622 
-49647 
-49672 
-49697 
.49723 

-86820 
-86805 
-86791 
-86777 
-86762 

-57155 
-57193 
-57232 
-57271 
.57309 

1-74964 
1-74846 
1-74728 
1-74610 
1  .  74492 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

.48226 
.48252 
48277 
.48303 
.48328 

•87603 
•87589 
.87575 
•87561 
•87546 

•55051 
.55089 
.55127 
.55165 
.55203 

1-81649 
1-81524 
1-81399 
1-81274 
1.81150 

-49748 
-49773 
-49798 
.49824 
.49849 

.86748 
.86733 
.86719 
.86704 
.86690 

-57348 
-57386 
-57425 
.57464 
.57503 

1-74375 
1-74257 
1-74140 
1  -  74022 
1-73905 

10 

9 
8 
7 

6 

55 
56 
57 
58 
59 

.48354 
.48379 
.48405 
.48430 
.48456 

•87532 
•87518 
.87504 
.87490 
.87476 

.55241 
.55279 
.55317 
.55355 
.55393 

1-81025 
1-80901 
1-80777 
1-80653 
i-80529 

-49874 
.49899 
.49924 
.49950 
.49975 

.86675 
-86661 
-86646 
-86632 
.86617 

-57541 
-57580 
-57619 
-57657 
.57696 

1-73788 
1-73671 
1-73555 
1-73438 
1-73321 

5 

4 
3 
2 
1 

60 

.  48481 

•87462 

•55431 

1-80405 

.50000 

-86603 

.57735 

1-73205 

O 

/ 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

' 

61° 


220 


60° 


TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS 
30°  31° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

f 

0 

1 

2 
3 

4 

.  50000 
.50025 
.50050 
.50076 
.50101 

86603 
86588 
86573 
.86559 
.86544 

.57735 
.57774 
.57813 
•57851 
•57890 

1.73205 
1.73089 
1.72973 
1.72857 
1-72741 

•51504 
•51529 
.51554 
.51579 
.51604 

.85717 
.85702 
.85687 
.85672 
.85657 

-60086 
-60126 
.60165 
.60205 
•60245 

1.66428 
1-66318 
1.66209 
1-66099 
1.65990 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.50126 
.50151 
.50176 
.50201 
.50227 

.86530 
.86515 
.86501 
.86486 
.86471 

.57929 
.57968 
•58007 
.58046 
.58085 

1-72625 
1-72509 
1-72393 
1-72278 
1-72163 

.51628 
•51653 
-51678 
.51703 
.51728 

•85642 
•85627 
.85612 
•85597 
•85582 

.60284 
-60324 
.60364 
.60403 
.  60443 

1.65881 
1-65772 
1-65663 
1-65554 
1.65445 

55 

54 
53 
52 
51 

10 

11 
12 
13 

.50252 
.50277 
-50302 
.50327 
.50352 

•86457 
•  86442 
•86427 
.86413 
•86398 

•58124 
.58162 
.58201 
.58240 
-58279 

1  .  72047 
1-71932 
1-71817 
1-71702 
1-71588 

•51753 
•51778 
•51803 
•51828 
.51852 

•85567 
•85551 
.85536 
.85521 
-85506 

-60483 
.60522 
•60562 
•60602 
.60642 

1-65337 
1.65228 
1-65120 
1-65011 
1  •  64903 

5O 

49 
48 
47 
48 

15 
16 
17 
18 
19 

•50377 
.  50403 
.50428 
.50453 
.50478 

•86384 
•86369 
•86354 
•86340 
-86325 

.58318 
.58357 
.58396 
.58435 
•58474 

1.71473 
1.71358 
1.71244 
1.71129 
1-71015 

.51877 
.51902 
.51927 
.51952 
-51977 

.85491 
.85476 
.85461 
.85446 
.85431 

.60681 
.60721 
.60761 
.60801 
.60841 

1.64795 
1-64687 
1-64579 
1-64471 
1.64363 

45 
44 
43 
42 
41 

20 

21 

22 
23 
24 

•50503 
.50528 
.50553 
.50578 
.50603 

•86310 
•86295 
.86281 
•86266 
•86251 

.58513 

-58552 
-58591 
-58631 
.58670 

1-70901 
1-70787 
1.70673 
1-70560 
1  -  70446 

•52002 
-52026 
-52051 
-52076 
•52101 

•85416 
.85401 
•85385 
.85370 
-85355 

.60881 
.60921 
.60960 
.61000 
•61040 

1-64256 
1.64148 
1-64041 
1.63934 
1.63826 

40 

39 
38 
37 
36 

25 
23 
27 
28 
39 

•50628 
•50654 
.50679 
.50704 
-50729 

.86237 
.86222 
.86207 
.86392 
.86178 

58709 
.58748 
.58787 
.58826 
.58865 

1-70332 
1.70219 
1.70106 
1-69992 
1-69879 

•52126 
.52151 
.52175 
.52200 
.52225 

-85340 
-85325 
-85310 
•85294 
.85279 

-61080 
-61120 
.61160 
-61200 
-61240 

1.63719 
1-63612 
1.63505 
1-63398 
1.63292 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34_ 

.50754 
.50779 
.50804 
•50829 
•50854 

.86163 
•86148 
.86133 
.86119 
.86104 

.58905 
.58944 
.58983 
.59022 
.59061 

1-69766 
1.69653 
1-69541 
1.69428 
1.69316 

•52250 
.52275 
-52299 
.52324 
-52349 

.85264 
•85249 
.85234 
.85218 
.85203 

.61280 
•61320 
-61360 
-61400 
.61440 

1-63185 
1.63079 
1.62972 
1-62866 
1.62760 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.50879 
•50904 
-50929 
•50954 
•50979 

•86089 
.86074 
•86059 
•86045 
•86030 

.59101 
.59140 
.59179 
59218 
.59258 

1-69203 
1-69091 
1-68979 
1.68866 

1-68754 

•52374 
•52399 
•52423 
•52448 
.52473 

.85188 
.85173 
•85157 
•85142 
-85127 

.61480 
•61520 
•61561 
.61601 
•61641 

1.62654 
1-62548 
1.62442 
1.62336 

1-62230 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

•51004 
.51029 
.51054 
•51079 
•51104 

•86015 
.86000 
85985 
•85970 
.85956 

.59297 
.59336 
.59376 
.59415 
.59454 

1.68643 
1-68531 
1.68419 
1.68308 
1.68196 

•52498 
•52522 
•52547 
•52572 
-52597 

.85112 
.85096 
•85081 
.85066 
•85051 

.61681 
.61721 
.61761 
.61801 
•61842 

1.62125 
].  62019 
1.61914 
1.61808 
1.61703 

20 

19 
18 
17 

16 

45 
46 
47 
48 
49 

.51129 
•51154 
.51179 
•51204 
•51229 

.85941 
.85926 
•85911 
.85896 
.85881 

.59494 
.59533 
.59573 
•59612 
.59651 

1-68085 
1.67974 
1-67863 
1-67752 
1-67641 

•52621 
•52646 
•52671 
.52696 
•52720 

•85035 
•85020 
.85005 
.84989 

.84974 

.61882 
•61922 
-61962 
•62003 
-62043 

1.61598 
1.61493 
1.61388 
1.61283 
1.61179 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

.51254 
•51279 
•51304 
•51329 
•51354 

.85866 
.85851 
<85836 
•85821 
•8o806 

•59691 
•59730 
•59770 
.59809 
•59849 

1-67530 
1.67419 
1.67309 
1-67198 
1.67088 

•52745 
•52770 
•52794 
•52319 
.52844 

•84959 
-  84943 
.84928 
.84913 
.84897 

•62083 
•62124 
.62164 
.62204 
•62245 

1-61074 
1.60970 
1  60865 
1-60761 
1.60657 

1O 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•51379 
51404 
.51429 
•51454 
•51479 

•85792 
.85777 
•85762 
•85747 
.85732 

•59888 
.59928 
•59967 
•60007 
.  60046 

1.66978 
1.66867 
1.66757 
1.66647 
1.66538 

-52869 
-52893 
-52918 
•52943 
.52967 

.84882 
.84866 
.84851 
.84836 
•84820 

.62285 
•62325 
.62366 
.62406 
.62446 

1.60553 
1-60449 
1-60345 
1  •  60241 
1-60137 

5 
4 
3 
2 
1 

?0 

•51504 

-85717 

•60086 

1-66428 

•52992 

.84805 

•62487 

1  -  60033 

O 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

59° 


221 


58° 


TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS.  AND  COTANGENTS 
33°  33° 


t 

Sin. 

Cos 

Tan. 

Cot 

Sin. 

Cos. 

Tan. 

Cot. 

< 

0 

2 
3 

.52992 
.53017 
.53041 
.53066 
.53091 

•84805 
.84789 
.84774 
.84759 
.  84743 

•62487 
.62527 
.62568 
.62608 
•62649 

1-60033 
1-59930 
1.59826 
1.59723 
1-59620 

-54464 
•54488 
•54513 
•54537 
.54561 

.83867 
•83851 
.83835 
•83819 
•83804 

•64941 
•64982 
•65024 
•65065 
•65106 

1.53986 
1.53888 
1.53791 
1-53693 
1-53595 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.53115 
.53140 
.53164 
.53189 
.53214 

•84728 

.84712 
•84697 
.84681 
.84666 

.62689 
.62730 
.62770 
•62811 
.62852 

1-59517 
1.59414 
1.59311 
1-59208 
1.59105 

.  54585 
-&4610 
.54635 
•54659 
.54683 

•83788 
•83772 
•83756 
•83740 
•83724 

-65148 
•65189 
•65231 
•65272 
•65314 

1.53497 
1  •  53400 
1-53302 
1-53205 
1.53107 

55 
54 
53 
52 
51. 

10 

11 
12 
13 
!!_ 

.53238 

53263 
53288 
»  53312 
.53337 

.84650 
.84635 
.84619 
.84604 
.84588 

•62892 
.62933 
.62973 
.63014 
.63055 

1.59002 
1-58900 
1.58797 
1.58695 
1.58593 

•54708 
•54732 
•54756 
•54781 
•54805 

.83708 
•83692 
•83676 
.83660 
.83645 

•65355 
•65397 
.65438 
•65480 
-65521 

1.53010 
1.52913 
1.52816 
1.52719 
1-52622 

50 

49 
48 
47 

46 

15 
16 
17 
18 
19 

.53361 
.53386 
.53411 
.53435 
.53460 

.84573 
.84557 
.84542 
.84526 
.84511 

.63095 
.63136 
.63177 
.63217 
.63258 

1.58490 
1.58388 
1-58286 
1.58184 
1.58083 

-54829 
.54854 
•54878 
.54902 
.54927 

.83629 
.83613 
.83597 
.83581 

.83565 

-65563 
-65604 
-65646 
•65688 
.65729 

1-52525 
1.52429 
1-52332 
1.52235 
1.52139 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

53484 
.53509 
•53534 
.53558 
.53583 

.84495 
.84480 
.84464 
.  84448 
.84433 

.63299 
.63340 
.63380 
.63421 
.63462 

1.57981 
1.57879 
1.57778 
1.57676 
1.57575 

54951 
.54975 
•54999 
•55024 
.55048 

.83549 
.83533 
.83517 
.83501 
.83485 

-65771 
-65813 
-65854 
-65896 
-65938 

1.52043 
1.51946 
1.51850 
1.51754 
1-51658 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.53607 
.53632 
.53656 
-53681 
.53705 

.84417 
.84402 
•84386 
.84370 
.84355 

•63503 
•63544 
.63584 
•63625 
.63666 

1.57474 
1.57372 
1.57271 
1.57170 
1.57069 

.55072 
.55097 
.55121 
•55145 
.55169 

.83469 
•83453 
.83437 
-83421 
.83405 

.65980 
-66021 
-66063 
-66105 
-66147 

1-51562 
1.51466 
1.51370 
1.51275 
1.51179 

35 
34 
33 
32 
31 

30 

81 
82 
33 
34 

.53730 
.53754 
•53779 
.53804 
.53828 

•84339 
.84324 
.84308 
.84292 
.84277 

.63707 
.63748 
.63789 
.63830 
.63871 

1.56969 
1.56868 
1.56767 
1.56667 
1.56566 

.55194 
.55218 
.55242 
.55266 
.55291 

.83389 
•83373 
•83356 
•83340 
.83324 

-66189 
.66230 
.66272 
.66314 
•66356 

1.51084 
1.50988 
1.50893 
1-50797 
1  .  50702 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.53853 
.53877 

.53902 
.53926 
.53951 

.84261 

.84245 
.84230 
.84214 
.84198 

.63912 
.63953 
.63994 
.64035 
.64076 

1.56466 
1.56366 
1.56265 
1.56165 
1.56065 

•55315 
.55339 
.55363 
.55388 

.55412 

.83308 
.83292 
.83276 
.83260 
•83244 

•66398 
•66440 
.66482 
.66524 
•66566 

1-50607 
1.50512 
1-50417 
1.50322 
1-50228 

25 
24 
23 
22 
21 

40 

41 
42 
43 

44 

.53975 
.54000 
.  54024 
.  54049 
.54073 

.84182 
.84167 
.84151 
•84135 
.84120 

.64117 
.64158 
•64199 
.  64240 
•64281 

1.55966 
1.55866 
1.55766 
1.55666 
1.55567 

.55436 
.55460 
•55484 
.55509 
.55533 

•83228 
.83212 
.83195 
.83179 
-83163 

.66608 
.66650 
.66692 
.66734 
.66776 

1.50133 
1.50038 
1.49944 
1.49849 
1.49755 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

•54097 
.54122 
•54146 
.54171 
.54195 

•84104 
•84038 
•84072 
•84057 
.  84041 

.64322 
•64363 
.  64404 
.  64446 
.64487 

1.55467 
1.55368 
.55269 
.55170 
.55071 

•55557 
.55581 
•55605 
•55630 
-55654 

-83147 
-83131 
-83115 
•83098 
•83082 

.66818 
.66860 
.66902 
.66944 
.66986 

1.49661 
1.49566 
1-49472 
1-49378 
1-49284 

15 
14 
13 
12 
11 

50 

51 
59, 
53 

54 

.54220 
.  54244 
.54269 
.54293 
.54317 

•84025 
.84009 
•83994 
.83978 
•83962 

.64528 
•64569 
•64610 
.64652 
.  64693 

•54972 
.54873 
.54774 
1-54675 
1-54676 

.55678 
.55702 
.55726 
•55750 
.55775 

•83066 
.83050 
.83034 
•83017 
.83001 

.67028 
.67071 
.67113 
.67155 
.67197 

1-49190 
1-49097 
1-49003 
1-48909 
1-48816 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

.54342 
•54366 
•  54391 
.54415 
.  54440 

.83946 
.83930 
•83915 
.83899 
•83883 

.64734 
.64775 
.64817 
•64858 
•64899 

1-54478 
1-54379 
1.54281 
1.54183 
1  •  54085 

•55799 
.55823 
.55847 
•55871 
.55895 

•82985 
82969 
•82953 
•82936 
•82920 

.67239 
.67282 
.67324 
.67366 
.67409 

1-48722 
1.48629 
1-48536 
1-48442 
1.48349 

5 

4 
3 
2 

60 

•  54464 

.83867 

•64941 

1-53986 

55919 

•82904 

.67451 

1.48256 

O 

' 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

57° 


222 


66° 


TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS, 
34°  35° 


Sin. 

Cos. 

Tan. 

Cot, 

Sin. 

Cos. 

Tan. 

Cot. 

9 

0 

2 
8 

55919 
55943 
55968 
55992 
56016 

82904 
82887 
82871 
82855 
82839 

67451 
67493 
67536 
67578 
67620 

1.48256 
1-48163 
1.48070 
1.47977 
1.47885 

57358 
57381 
57405 
57429 
57453 

81915 
81899 
81882 
81865 
81848 

70021 
70064 
.70107 
•70151 
•70194 

1.42815 
1.42726 
1.42638 
.42550 
.42462 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

56040 
56064 
56088 
56112 
56136 

82822 
82806 
82790 
82773 
82757 

67663 
67705 
67748 
67790 
67832 

1-47792 
1-47699 
1.47607 
1.47514 
1.47422 

57477 
57501 
57524 
57548 
.57572 

81832 

81815 
81798 
.81782 
•81765 

•70238 
•70281 
•70325 
•70368 
-70412 

.42374 
.42286 
•42198 
.42110 
1-42022 

53 
54 
53 
52 
51 

10 

11 
12 
13 
14 

56160 
56184 
56208 
56232 
56256 

82741 
82724 
82708 
.82692 
.82675 

67875 
67917 
67960 
68002 
68045 

1.47330 
1.47238 
1.47146 
1-47053 
1-46962 

•57596 
.57619 
•57643 
.57667 
.57691 

•81748 
•81731 
•81714 
•81698 
.81681 

•70455 
•70499 
-  70542 
-70586 
-70629 

1-41934 
1.41847 
1.41759 
1.41672 
1.41584 

5O 

49 
48 
47 
46 

15 
16 
17 
18 
19 

56280 
56305 
56329 
56353 
56377 

.82659 
.82643 
.82626 
.82610 

.82593 

68088 
68130 
.68173 
.68215 

.68258 

1.46870 
1.46778 
1.46686 
1.46595 
1.46503 

-57715 
.57738 
-57762 
•57786 
.57810 

-81664 
.81647 
.81631 
.81614 
•81597 

-70673 
-70717 
-70760 
-70804 
-  70848 

1.41497 
1.41409 
1-41322 
1.41235 
1.41148 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

56401 
.56425 
.  56449 
.  56473 
.56497 

.82577 
.82561 
.82544 
.82528 
.82511 

.68301 
.68343 
.68386 
.68429 
.68471 

1.46411 
1-46320 
1-46229 
1.4*6137 
1.46046 

.57833 
•57857 
•57881 
•57904 
.57928 

-81580 
.81563 
.81546 
.81530 
-81513 

•70891 
•70935 
-70979 
•71023 
•71066 

1.41061 
1.40974 
1.40887 
1.40800 
1-40714 

4O 

39 
38 
37 

36 

25 
26 
27 
28 
29 

.56521 
56545 
.56569 
.56593 
.56617 

•82495 
.82478 
.82462 
.82446 
.82429 

.68514 
.68557 
.68600 
.68642 
.68685 

1.45955 
1.45864 
1.45773 
1.45682 
1.45592 

.57952 
-57976 
.57999 
-58023 
.58047 

-81496 
•81479 
.81462 
.81445 
.81428 

.71110 
•71154 
-71198 
.71242 
-71285 

•40627 
.40540 
.40454 
.40367 
.40281 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.56641 
.56665 
.56689 
.56713 
•56736 

.82413 
.82396 
.82380 
.82363 
.82347 

.68728 
.68771 
.68814 
.68857 
.68900 

1.45501 
1.45410 
1.45320 
1.45229 
1.45139 

.58070 

.58094 
.58118 
.58141 
.58165 

.81412 
.81395 
.81378 
.81361 
.81344 

•71329 
•71373 
.71417 
.71461 
.71505 

.40195 
.40109 
1-40022 
1.39936 
1-39850 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.56760 
.  5678<c 
.56803 
.56832 
-568'i6_ 

.82330 
.82314 
.82297 
.82281 

.82264 

.68942 
68985 
.69028 
.69071 

-69114 

1-45049 
1-44958 
1.44868 
1-44778 
1-44688 

.58189 
.58212 
.58236 
.58260 
-58283 

•81327 
.81310 
.81293 
.81276 

.81259 

.71549 
.71593 
.71637 
.71681 

•71725 

1-39764 
1.39679 
1.39503 
1-39507 

1-39421 

25 
24 
23 
22 
-21 

40 

41 
42 
43 
44 

.56880 
.56904 
.56928 
•56952 
.56976 

.82248 
•82231 
.82214 
.82198 
.82181 

.69157 
.69200 
.69243 
.69286 
-69329 

1-44598 
1.44508 
1.44418 
1-44329 
1.44239 

-58307 
-58330 
-58354 
-58378 
.58401 

•81242 
•81225 
•81208 
•81191 
•81174 

-71769 
-71813 
.71857 
.71901 
.71946 

1-39336 
1-39250 
1-39165 
1.39079 
1-38994 

20 

19 
18 
17 
16 

45 
46 
47 

48 
49 

.57000 
.57024 
•57047 
.57071 
.57095 

.82165 
.82148 
•82132 
.82115 
.82098 

•69372 
.69416 
•69459 
.69502 
.69545 

1.44149 
1.44060 
1-43970 
1.43881 
1-43792 

.58425 
.  58449 
.58472 
.58496 
.58519 

.81157 
•81140 
.81123 
•81106 
.81089 

-71990 
-72034 
.72078 
.72122 
•72167 

1-38909 
1-38824 
1.38738 
1-38653 
1.38568 

15 
14 
13 
12 
Ui 

50 

51 
52 
53 
54 

.57119 
.57143 
.57167 
•57191 
.57215 

•82082 

•82065 
•82048 
.82032 
.82015 

•69588 
•69631 
•69675 
•69718 
•69761 

1.43703 
1-43614 
1.43525 
1-43436 
1.43347 

.58543 
.58567 
.58590 
.58614 
•58637 

•81072 
•81055 
.81038 
-81021 
.81004 

•72211 
•72255 
.72299 
.  72344 
•72388 

1.38484 
1.38399 
1-38314 
1-38229 
1.38145 

10 
9 
8 
7 
_6 

55 
56 
57 
58 

59 

•57238 
•57262 
•57286 
•57310 
•57334 

.81999 
•81982 
•81965 
81949 
81932 

•69804 
•69847 
•69891 
•69934 
•69977 

1-43258 
1.43169 
1-43080 
1.42992 
1.42903 

.58661 
.58684 
•58708 
.58731 
.58755 

-80987 
.80970 
.80953 
.80936 
.80919 

.72432 
•72477 
.72521 
.72565 
-72610 

1.38060 
1.37976 
1.37891 
1.37807 
1-37722 

9 

3 

2 

60 

.57358 

81915 

.70021 

1-42815 

.58779 

-80902 

-72654 

1-37638 

0. 

"^ 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

£ 

;s° 

2 

23 

fi 

4° 

TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS 
36°  37° 


Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

, 

0 

2 
3 

•58779 
•58802 
58826 
.58849 
.58873 

80902 
80835 
80867 
80350 
80833 

.72654 
.72699 
.72743 
.72788 
.72832 

1-37638 
1.37554 
1.37470 
1.37386 
1-37302 

•60182 
•60205 
•60228 
•60251 
.  60274 

-79864 
-  79846 
-79829 
.79811 
-79793 

-75355 
.  75401 
•75447 
-75492 
-75538 

1-32704 
1-32624 
1-32544 
1-32464 
1-32384 

60 

59 
58 
57 

56 

5 
6 
7 
8 
9 

58896 
58920 
58943 
58967 
58990 

80816 
80799 
80782 
80765 
80748 

•72877 
.72921 
72966 
.73010 
73055 

1.37218 
1.37134 
1.37050 
1.36967 
1.36883 

.  60298 
.60321 
.60344 
-60367 
.60390 

.79776 
.79758 
.79741 
.79723 
.79706 

-75584 
-75629 
-75675 
.75721 
.75767 

1-32304 
1-32224 
1-32144 
1-32064 
1.31984 

55 
54 
53 
52 
SI 

10 

11 
12 
13 
14 

59014 
59037 
59061 
59084 
59108 

80730 
80713 
80696 
80679 

80662 

73100 
73144 
73189 
73234 
73278 

1.36800 
1.36716 
1.36633 
1.36549 
1.36466 

.  60414 
.60437 
.  60460 
.60483 
-60506 

.79688 
.79671 
.79653 
.79635 
.79618 

.75812 
.75858 
-75904 
•75950 
.75996 

1.31904 
1.31825 
1-31745 
1.31666 
1-31586 

50 

49 
48 
47 
46 

15 
16 
17 
18 

19 

59131 
59154 
59178 
59201 
59225 

80644 
80627 
80610 
80593 
80576 

73323 
73368 
73413 
73457 
73502 

1.36383 
1.36300 
1.36217 
1.36134 
1.36053 

•60529 
.60553 
.60576 
.60599 
.60622 

•79600 
•79583 
.79565 
-79547 
.79530 

•76042 
.76088 
.76134 
•76180 
76226 

1-31507 
1.31427 
1-31348 
1-31269 
1-31190 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

59248 
59272 
59295 
59318 
59342 

80558 
8054.1 
80524 
80507 
80489 

73547 
73592 
73637 
73681 
73726 

1.35968 
1.35885 
1.35802 
1.35719 
1-35637 

.  60645 
.60668 
•60691 
-60714 
•60738 

79512 
.79494 
.79477 
•79459 
•79441 

.76272 
.76318 
.76364 
•76410 
.76456 

1-31110 
1.31031 
1.30952 
1.30873 
1-30795 

4O 

39 
38 
37 
36 

25 
26 
27 
28 
29 

59365 
59389 
59412 
59436 
59459 

80472 
80455 
80438 
80420 
80403 

73771 
73816 
73861 
73906 
73951 

1.35554 
1-35472 
1.35389 
1.35307 
1.35224 

.60761 
•6G784 
•60807 
•60830 
•60853 

.79424 
•79406 
•79388 
•79371 
•79353 

.76502 
.76548 
.76594 
-76640 
-76686 

1-30716 
1.30637 
1.30558 
1  •  30480 
1-30401 

35 
34 
33 
32 
31 

30 

81 
32 
33 
34 

59482 
59506 
59529 
59552 
59576 

80386 
80368 
80351 
80334 
80316 

73996 
.74041 
74086 
74131 
74176 

1.35142 
1.35060 
1.34978 
1.34896 
1.34814 

•60876 
•60899 
•60922 
•60945 
.60968 

.79335 
•79318 
.79300 
.79282 
.79264 

.76733 
.76779 
-76825 
-76871 
.76918 

1-30323 
1.30244 
1-30166 
1-30087 
1-30009 

30 

29 
28 
27 
26 

35 
36 
37 
38 

39_ 

59599 
59622 
59646 
59669 
.  59693^ 

80299 
80282 
80264 
80247 
80230 

74221 
74267 
74312 
74357 
.74402 

1.34732 
1.34650 
1-34568 
1.34487 

1-34405 

.60991 
.61015 
•61038 
•61061 
•61084 

79247 
.79229 
-79211 
.79193 
.79176 

.76964 
.77010 
•77057 
-77103 
•77149 

1-29931 
1-29853 
1.29775 
1-29696 
1-29318 

25 
24 
23 
22 
21 

40 

41 
42 
43 

44 

.59716 
.59739 
.59763 
.59786 
.59809 

80212 
80195 
.80178 
.80160 
.80143 

•74447 
.74492 
.74538 
.74583 
.74628 

1.34323 
1-34242 
1.34160 
1.34079 
1.33998 

•61107 
.61130 
•61153 
•81176 
•61199 

-79158 
-79140 
.79122 
.79105 
.79087 

•77196 
-77242 
•77289 
•77335 
.77382 

1-29541 
1-29463 
1-29385 
1.29307 
1-29229 

20 

19 
18 
17 
16 

45 
46 
47 
48 
40 

59832 
.59856 
.59879 
.59902 
.59926 

.80125 
•80108 
.80091 
.80073 
.80056 

.74674 
.74719 
.74764 
.74810 
•74855 

1.33916 
1.33835 
1.33754 
1.33673 
1.33592 

•61222 
•61245 
•61268 
.61291 
•61314 

.79069 
.79051 
.79033 
.79016 
.78998 

-  77428 
-77475 
-77521 
-77568 
-77615 

•29152 
•29074 
•28997 
.28919 
.28842 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

.59949 
.59972 
•59995 
•60019 
.60042 

.80038 
.80021 
.80003 
.79986 
.79968 

.74900 
.74946 
.74991 
.75037 
.75082 

1.33511 
1.33430 
1.33349 
1.33268 
1.33187 

•61337 
•61360 
161383 
j.  61406 
.61429 

.78980 
.78962 
.78944 
.78926 
.78908 

-77661 
.77708 
.77754 
.77801 
.77848 

.28764 
•28687 
.28610 
.28533 
.28456 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•60065 
.60089 
•60112 
•60135 
.60158 

.79951 
.79934 
.79916 
.79899 
.79881 

.75128 
•75173 
.75219 
.75264 
.75310 

1.33107 
1.33026 
1-32946 
1.32865 
1.32785 

•61451 
.61474 
.61497 
•61520 
.61543 

-78891 
•78873 
.78855 
-78837 
78819 

.77895 
.77941 
-77988 
-78035 
-78082 

-28379 
-28302 
-28225 
-28148 
1.28071 

5 
4 
3 
2 

60 

.60182 

.79864 

.75355 

1.32704 

.61566 

•78801 

-78129 

1-27994 

0 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

'- 

I 

a* 

2'< 

VL 

i 

»• 

-NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
38°  39° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

1 

2 
3 
4 

.61566 
•61589 
•61612 
•61635 
•61658 

•78801 
•78783 
.78765 
•78747 
•78729 

•78129 
.78175 
•78222 
-78269 
•78316 

1-27994 
1.27917 
1.27841 
1.27764 
1-27688 

•62932 
.62955 
.62977 
•63000 
•63022 

•77715 
.77696 
.77678 
.77660 
.  77641 

.80978 
•81027 
.81075 
•81123 
•81171 

1-23490 
1-23416 
1-23343 
1.23270 
1.23196 

6O 

59 
58 
57 
56 

5 
6 
7 
8 
9 

•61681 
.61704 
•61726 
.61749 
•61772 

.78711 
.78694 
•78676 
78658 
.  78640 

.78363 
•78410 
.  78457 
•78504 
•78551 

1.27611 
1.27535 
1.27458 
1-27382 
1.27306 

.  63045 
-63068 
•63090 
•63113 
.63135 

•77623 
-77605 
77586 
-77568 
-77550 

81220 
.81268 
.81316 
.81364 
•81413 

1.23123 
1.23050 
1-22977 
1.22904 
1-22831 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

•61795 
.61818 
.61841 
•61864 
.61887 

.78622 
•78604 
.78586 
•78568 
-78550 

.78598 
78645 
•78692 
.78739 
•78786 

1.27230 
1.27153 
1.27077 
1.27001 
1.26925 

.63158 
•63180 
.63203 
•63225 
•63248 

.77531 
.77513 
.  77494 
•77476 
•77458 

•81461 
•81510 
.81558 
•81606 
.81655 

1.22758 
1.22685 
1-22612 
1-22539 
1-22467 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

•61909 
•61932 
.61955 
•61978 
•62001 

.78532 
•78514 
•78496 
•78478 
-78460 

•78834 
.78881 
.78928 
-78975 
-79022 

1.26849 
1.26774 
1.26698 
1.26622 
1.26546 

•63271 
•63293 
•63316 
-63338 

-63361 

•77439 
.  77421 
.  77402 
.77384 
•77366 

.81703 
.81752 
•81800 
•81849 
-81898 

1-22394 
1.22321 
1.22249 
1-22176 
1.22104 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

•62024 
.62046 
•62069 
.62092 
•62115 

.  78442 
•78424 
•78405 
•78387 
.78369 

.79070 
.79117 
•79164 
.79212 
•79259 

1-26471 
1-26395 
1.26319 
1-26244 
1.26169 

-63383 

-63406 
-63428 
-63451 
-63473 

•77347 
.77329 
.77310 
.77292 
•77273 

-81946 
-81995 
-82044 
•82092 
.82141 

1-22031 
1-21959 
1-21886 
1-21814 
1.21742 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

•62138 
•62160 
•62183 
•62206 
•62229 

•78351 
.78333 
•78315 
.78297 
•78279 

•79306 
.79354 
.  79401 
.  79449 
.79496 

1-26093 
1-26018 
1-25943 
1-25867 
1-25792 

-63496 
-63518 
-63540 
-63563 
.63585 

•77255 
•77236 
-77218 
.77199 
•77181 

•82190 
•82238 
•82287 
.82336 
.82385 

1-21670 
1.21598 
1.21526 
1.21454 
1-21382 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

•62251 

•62274 
.62297 
.62320 
.62342 

•78261 
.78243 
•78225 
•78206 
.78188 

•79544 
.79591 
.79639 
•79686 
.79734 

1.25717 
1-25642 
1.25567 
1-25492 
1.25417 

•63608 
.63630 
.63653 
•63675 
-63698 

-77162 

.  77144 
-77125 
•77107 
•77088 

.  82434 
•82483 
.82531 
-82580 
-82629 

1.21310 
1-21238 
1-21166 
1-21094 
1-21023 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

•62365 
.62388 
.62411 
.62433 
•62456 

•78170 
.78152 
•78134 
.78116 
.78098 

.79781 
.79829 
.79877 
-79924 
•79972 

1.25343 
1.25268 
1.25193 
1.25118 

1.25044 

.63720 
.63742 
-63765 
-63787 
-63810 

•77070 
•77051 
•77033 
.77014 
•76996 

-82678 
•82727 
•82776 
•82825 
.82874 

1-20951 
1-20879 
1-20808 
1-20736 

1-20665 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

•62479 
.62502 
-62524 
•62547 
.62570 

•78079 
.78061 
.78043 
•78025 
-78007 

.80020 
•80067 
.80115 
•80163 
.80211 

1.24969 
1.24895 
1-24820 
1.24746 
1.24672 

•63832 
.63854 
•63877 
•63899 
.63922 

•76977 
•76959 
.  76940 
•76921 
•76903 

.82923 
.82972 
•83022 
.83071 
.83120 

1-20593 
-20522 
•20451 
-20379 
•20308 

2O 

19 
18 
17 
16 

45 
46 
47 
48 
49 

.62592 
•62615 
•62638 
•62660 
•62683 

.77988 
.77970 
.77952 
.77934 
.77916 

•80258 
•80306 
•80354 
.80402 
•80450 

1.24597 
1.24523 
1.24449 
1.24375 
1.24301 

•63944 
•63966 
-63989 
•64011 
.64033 

•76884 
•76866 
•76847 
•76828 
•76810 

.83169 
.83218 
•83268 
•83317 
-83366 

.20237 
•20166 
•20095 
1-20024 
1-19953 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

•62706 
•62728 
•62751 
•62774 
.62796 

•77897 
.77879 
.77861 
.  77843 
.77824 

•80498 
.  80548 
-80594 
•80642 
.80690 

1-24227 
1.24153 
1.24079 
1.24005 
1.23931 

•64056 
•64078 
.  64100 
.64123 
•64145 

-76791 
-76772 
-76754 
.76735 
•76717 

-83415 
-83465 
-83514 
-83564 
-83613 

1-19882 
1-19811 
1-19740 
1-19669 
1-19599 

10 

9 
8 

7 
6 

55 
56 
57 
58 
59 

•62819 
•62842 
•62864 
•62887 
.62909 

•77806 
-77788 
.77769 
•77751 
.77733 

•80738 
.80786 
.80834 
•80882 
•80930 

1.23858 
1-23784 
1.23710 
1-23637 
1-23563 

.64167 
•64190 
.64212 
.64234 
•64256 

-76698 
•76679 
•76661 
•76642 
•76623 

-83662 
-83712 
-83761 
-83811 
-83860 

1-19528 
1.19457 
1.19387 
1.19316 
1-19246 

5 

4 
3 
2 
1 

60 

•62932 

•77715 

80978 

1-23490 

-64279 

-76604 

-83910 

1.19175 

0 

/ 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

51° 


225 


50° 


TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 
40°  41° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

t 

0 

2 
3 
4 

64279 
64301 
64323 
64346 
64368 

76604 
76586 
76567 
76548 
76530 

83910 
83960 
84009 
84059 
84108 

1.19175 
1-19105 
1-19035 
1-18964 
1-18894 

65606 
65628 
65650 
65672 
65694 

75471 
75452 
75433 
75414 
75395 

86929 
86980 
87031 
87082 
87133 

1.15037 
1.14969 
1-14902 
1-14834 
1-14767 

60 

69 
58 
57 
56 

5 
6 
7 
8 
9 

64390 
64412 
64435 
64457 
64479 

76511 
76492 
76473 
76455 
76436 

84158 

84208 
84258 
84307 
84357 

1.18824 
1-18754 
1.18684 
1-18614 
1.18544 

65716 
65738 
65759 
65781 
65803 

75375 
75356 
75337 
75318 
•75299 

87184 
87236 
87287 
87338 
87389 

1-14699 
1-14632 
1-14565 
1.14498 
1.14430 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

64501 
64524 
64546 
64568 
64590 

76417 
76398 
76380 
76361 
76342 

84407 
84457 
84507 
84556 
84606 

1.18474 
1  .  18404 
1.18334 
1-18264 
1.18194 

65825 
65847 
.65869 
.65891 
•65913 

.75280 
•75261 
.  75241 
•75222 
.75203 

87441 
87492 
87543 
87595 
87646 

1-14363 
1-14296 
1-14229 
1.14162 
1.14095 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19  , 

64612 
64635 
64657 
64679 
64701 

76323 
76304 
•76286 
.76267 
76248 

84656 
84706 
•84756 
84806 
84856 

1.18125 
1.18055 
1-17986 
1-17916 
1-17846 

.65935 
.65956 
•65978 
-  66000 
-66022 

•75184 
•75165 
•75146 
•75126 
•75107 

.87698 
•87749 
•87801 
•87852 
.87904 

1-14028 
1-13961 
1.13894 
1-13828 

1-13761 

45 
44 
43 
42 
41 

2O 

21 
22 
23 
24 

64723 
.  64746 
.64768 
•64790 
•64812 

.76229 
•76210 
•76192 
•76173 
•76154 

•84906 
•  ii4956 
•85006 
•85057 
•85107 

1.17777 
1-17708 
1-17638 
1-17569 
1.17500 

.  66044 
-66066 
-66088 
-66109 
•66131 

.75088 

•75069 
-75050 
•75030 
•75011 

.87955 
.88007 
•88059 
•88110 
•88162 

1-13694 
1-13627 
1-13561 
1-13494 
1-13428 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.64834 
.64856 
•64878 
.64901 
.64923 

-76135 
•76116 
.76097 
.76078 
.76059 

•85157 
•85207 
•85257 
•85308 
•85358 

1-17430 
1.17361 
1-17292 
1.17223 
1-17154 

•66053 
•66175 
•66197 
•66218 
.  66240 

.74992 
•74973 
•74953 
.  74934 
•74915 

•88204 
•88265 
•88317 
•88369 
•88421 

1.13361 
1-13295 
1-13228 
1-13162 
1-13096 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.  64945 
•64967 
•64989 
.65011 
.65033 

•76041 
.76022 
.76003 
•75984 
•75965 

•85408 
•85458 
•85509 
•85559 
•85609 

1-17085 
1-17016 
1-16947 
1.16878 
1.16809 

•66262 
.66284 
•66306 
•66327 
•66349 

•74896 
-74876 
-74857 
•74838 
•74818 

•88473 
•88524 
•88576 
-88628 
-88680 

1.13029 
1.12963 
1.12897 
1-12831 
1-12765 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.65055 
•65077 
.65100 
•65122 
•65144 

.75946 
•75927 
•75908 
•75889 
•75870 

•85660 
•85710 
•85761 
•85811 
-85862 

1-16741 
1.16672 
1-16603 
1-16535 
1-16466 

•66371 
•66393 
•66414 
•66436 

-66458 

•74799 
•  74780 
.  74760 
.  74741 

-74722 

•88732 
-88784 
-88836 
-88888 
-88940 

1-12699 
1-12633 
1-12567 
1-12501 
1-12435 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

•65166 
•65188 
•65210 
•65232 
•65254 

•75851 
•75832 
-75813 
•75794 
•75775 

-85912 
-85963 
•86014 
•86064 
-86115 

1.16398 
1.16329 
1-16261 
1-16192 
1.16124 

.  66480 
•66501 
.66523 
•66545 
•66566 

-  74703 
-74683 
.  74664 
•  74644 
-74625 

.88992 
•89045 
-89097 
•89149 
.89201 

1.12369 
1.12303 
1.12238 
1-12172 
1.12106 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

•65276 
•65298 
•65320 
•65342 
•65364 

.75756 
•75738 
•75719 
•75700 
•75680 

•86166 
86216 
.86267 
•86318 
-86368 

1.16056 
1-15987 
1-15919 
1-15851 
1-15783 

•66588 
•66610 
•66632 
•66653 
•66675 

.  74606 
•  74586 
•74567 
•  74548 
•  74528 

•89253 
.89306 
•89358 
-89410 
.89463 

1.12041 
1-11975 
1.11909 
1-11844 
1-11778 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

.65386 
.65408 
.65430 
•65452 
.65474 

•75661 
.75642 
•75623 
•75604 
•75585 

•86419 
•86470 
•86521 
•86572 
•86623 

1-15715 
1.15647 
1-15579 
1-15511 
1-15443 

•66697 
•66718 
•66740 
•66762 
•66783 

.  74509 
.  74489 
.  74470 
.  74451 
.74431 

•89515 
•89567 
•89620 
•89672 
•89725 

1-11713 
1.11648 
1.11582 
1.11517 
1-11452 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•65496 
.65518 
•65540 
.65562 
•65584 

•75566 
•75547 
•75528 
.75509 
.75490 

•86674 
•86725 
.86776 
.86827 
.86878 

1.15375 
1.15308 
1-15240 
1-15172 
1-15104 

-66805 
•66827 
•66848 
•66870 
•66891 

.74412 
•74392 
•74373 
•74353 
.  74334 

.89777 
.89830 
•89883 
•89935 
•89988 

1-11387 
1-11321 
1-11256 
1-11191 
1.11126 

5 

3 

2 

60 

.65606 

.75471 

-86929 

1.15037 

-66913 

-74314 

-90040 

1-11061 

0 

' 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

' 

49° 


226 


48° 


-NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANuENTS. 
43°  43° 


t 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

t 

o 

2 
8 

.66913 
66935 
.66956 
.66978 
.66999 

•74314 

74295 
74276 
74256 
74237 

.90040 
•90093 
•90146 
•90199 
.90251 

1-11061 
1  10996 
1-10931 
1.10867 
1.10802 

68200 
68221 
68242 
68264 
68285 

73135 
-73116 
73096 
-73076 
-73056 

93252 
-93306 
-93360 
-93415 
-93469 

1-07237 
1-07174 
1-07112 
1-07049 
1-06987 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.67021 
67043 
.67064 
•67086 
.67107 

74217 
74198 
74178 
74159 
74139 

90304 
90357 
90410 
90463 
90516 

1-10737 
1.10672 
1-10607 
1-10543 
1  •  10478 

68306 
68327 
68349 
68370 
68391 

73036 
-73016 
72996 
72976 
-72.957 

-93524 
-93578 
•93633 
-93688 
-93742 

1-06925 
1-06862 
1-06800 
1-06738 
1-06676 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.67129 
.67151 
67172 
67194 
67215 

74120 
74100 
74080 
74061 
74041 

90569 
90621 
90674 
90727 
90781 

1-10414 
1-10349 
1-10285 
1-10220 
1.10156 

68412 
68434 
68455 
68476 
68497 

-72937 
72917 
72897 
72877 
72857 

-93797 
-93852 
-93906 
-93961 
-94016 

1-06613 
1-06551 
1-06489 
1-06427 
1  06365 

5O 

49 
48 
47 
46 

15 
16 
17 
18 

19 

67237 
67258 
67280 
67301 

87333 

74022 
74002 
73983 
73963 
73944 

90834 
90887 
90940 
90993 
91046 

1.10091 
1.10027 
1.09963 
1.09899 
1-09834 

68518 
68539 
68561 
68582 
68603 

72837 
72817 
72797 
72777 
72757 

.94071 
-94125 
-94180 
-94235 
-94290 

-06303 
-06241 
-06179 
-06117 

.06056 

45 
44 
43 
42 
.41 

20 

21 
22 
23 
24 

67344 
67366 
67387 
67409 
67430 

73924 
73904 
73885 
73865 
73846 

91099 
91153 
91206 
91259 
91313 

1.09770 
1-09706 
1.09642 
1-09578 
1-09514 

68624 
68645 
68666 
68688 
68709 

72737 
72717 
72697 
72677 
72657 

-94345 
-94400 
94455 
-94510 
-94565 

-05994 
-05932 
1-05870 
1-05809 
1-05747 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

67452 
67473 
67495 
67516 
67538 

73826 
73806 
73787 
73767 
73747 

91368 
91419 
91473 
91526 
91580 

1-09450 
1-09386 
1-09322 
1-09258 
1-09195 

68730 
68751 
68772 
68793 
68814 

72637 
72617 
72597 
72577 
72557 

.94620 
.94676 
.94731 
•94786 
.  94841 

1-05685 
1-05624 
1-05562 
1-05501 
1-05439 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

67559 
67580 
67602 
67623 
67645 

73728 
73703 
73688 
73669 
73849 

91633 
91687 
91740 
91794 
91847 

1-09131 
1-09067 
1-09003 
1-08940 
1-08876 

68835 
68857 
68378 
68899 
68920 

72537 
72517 
72497 
72477 
.72457 

•94896 
•94952 
•95007 
•95062 
.95118 

1-05378 
1-05317 
1-05255 
1-05194 
1-05133 

30 

29 
28 

le 

35 
36 
37 
38 

39 

•67666 
•67688 
-67709 
•67730 
67752 

73629 
73810 
73590 
73570 
73551 

91901 
91955 
92008 
92062 
92116 

1-08813 
1-08749 
1.08686 
1-08622 
1-08559 

68941 
68962 
68983 

69004 
69025 

.72437 
-72417 
-72397 
-72377 
-72357 

•95173 
•95229 
•95284 
•95340 
.95395 

1.05072 
1.05010 
1  .  04949 
1-04888 

1.04827 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

.67773 
•67795 
•67816 
•67837 
.67859 

73531 
73511 
73491 
73472 
73452 

92170 
92224 
92277 
92331 
92385 

1-08496 
1-08432 
1.08369 
1-08306 
1-08243 

69046 
69067 
69088 
69109 
.69130 

-72337 
-72317 
•72297 
.72277 
.72257 

.95451 
.95506 
.95562 
.95618 
.95673 

1-04766 
1-04705 
1.04644 
1-04583 
1-04522 

20 

19 
18 
17 
_16 

45 
46 
47 
48 
49 

-67880 
•67901 
.67923 
•67944 
•67965 

73432 
73413 
73393 
73373 
73353 

92439 
92493 
92547 
92601 
.92655 

1.08179 
1-08118 
1.08053 
1.07990 
1.07927 

-69151 
-69172 
-69193 
-69214 
.69235 

.72236 
.72216 
•72196 
•72176 
.72156 

.95729 
.95785 
-95841 
.95897 
.95952 

1-04461 
1-04401 
1-04340 
1  .  04279 
1-04218 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54_ 

.67987 
•68008 
•68029 
•68051 
•68072 

73333 
73314 
73294 
73274 
•73254 

•92709 
•92763 
.92817 
•92872 
.92926 

1-07864 
1-07801 
1-07738 
1-07676 
1-07613 

-69256 
•69277 
-69298 
.69319 
-  69340 

.72136 
.72116 
.72095 
.72075 
.72055 

.96008 
.96064 
.96120 
.96176 
.98232 

1.04158 
1.04097 
1-04038 
1.03976 
1-03915 

10 

9 

? 
_6 

55 
56 
57 
58 
59 

•68093 
•68115 
.68138 
•68157 
.68179 

•73234 
.73215 
.73195 
•73175 
•73155 

.92980 
•93034 
.93088 
•93143 
•93197 

1.07550 
1.07487 
1-07425 
1-07362 
1-07299 

-69361 
.69382 
.69403 
•69424 
.  69445 

•72035 
.72015 
.71995 
.71974 
.71954 

.96288 
-96344 
-96400 
96457 
.96513 

1-03855 
1.03794 
1.03734 
1-03674 
1.03613 

5 

3 
2 

60 

.68200 

.73135 

•93252 

1-07237 

69466 

71934 

-96569 

1.03553 

0 

—  7— 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

47° 


227 


46° 


TABLE  10.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS 
44°  44° 


/ 
0 

2 
3 
,4 
5 
6 
7 
8 
9 

Sin. 

Cos. 

Tan. 

Cot. 

' 

; 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

.69466 
•69487 
.69508 
•69529 
•69549 

•71934 
.71914 
•71894 
•71873 
•71853 

.96569 
.96625 
.96681 
.96738 
.96794 

1.03553 
1.03493 
1.03433 
1-03372 
1-03312 

60 

59 
58 
57 
56 

3O 
31 

32 
33 
34 

.70091 
.70112 
.70132 
•70153 
.70174 

•71325 
-71305 
-71284 
-71264 
-71243 

.98270 
-98327 
.98384 
-98441 
.98499 

1-01761 
1-01702 
1-01642 
1.01583 
1-01524 

30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
-IS 
15 
14 
13 
12 
11 

.69570 
•69591 
•69612 
•69633 
•69654 

•71833 
.71813 
.71792 
.71772 
.71752 

•96850 
•96907 
.96963 
.97020 
.97076 

1.03252 
1.03192 
1-03132 
1.03072 
1.03012 

55 
54 
53 
52 
51 

35 
36 
37 
38 

39 

•70195 
•70215 
•70236 
•70257 
.70277 

•71223 
-71203 
-71182 
-71162 
•71141 

.98556 
.98613 
•98671 
.98728 
.98786 

1-01465 
1.01406 
1.01347 
1.01288 
1-01229 

10 

11 
12 
13 
ii_ 
15 
16 
17 
18 

i£- 
20 

21 
22 
23 
24_ 
25 
26 
27 
28 
29_ 

&L 

•69675 
•69696 
.69717 
•69737 
.69758 
•69779 
.69800 
.69821 
.69842 
.69862 

•71732 
.71711 
.71691 
•71671 
.71650 

.97133 
.97189 
.97246 
•97302 
•97359 

1-02952 
1.02892 
1.02832 
1.02772 
1.02713 

50 

49 
48 
47 
46 

40 

41 
42 
43 
44 

-70298 
-70319 
.  70339 
.70360 
.70381 

-71121 
.71100 
.  71080 
.71059 
.71039 

98843 
.98901 
•98958 
.99016 
-99073 

1-01170 
1.01112 
1.01053 
1-00994 
1.00935 

.71630 
.71610 
.71590 
.71569 
•71549 

•97416 
.97472 
•97529 
.97586 
.97643 

1-02653 
1.02593 
1.02533 
1.02474 
1-02414 

45 
44 
43 
42 
41 

45 
46 
47 
48 
49 

•  70401 
-  70422 
70443 
-70463 
-  70484 

.  71019 
•70998 
-70978 
-70957 
70937 

-99131 
-99189 
-99247 
.99304 
-99362 

1.00876 
1.00818 
1-00759 
1.00701 
1.00642 

.69883 
•69904 
.69925 
.69946 
.69966 

.71529 
.71508 
•71488 
.71468 
•71447 

•  97700 
.97756 
•97813 
.97870 
.97927 

1-02355 
1.02295 
1.02236 
1.02176 
1.02117 
1-02057 
1.01998 
1-01939 
1-01879 
1.01820 

40 

39 
38 
37 
36 

50 

51 
52 
53 
54 

•70505 
•70525 
•70546 
.70567 
•70587 

•70916 
•70896 
•  70875 
•  70855 
•  70834 

.99420 
•99478 
-99536 
.99594 
.99652 

1.00583 
1.00525 
1-00467 
1-00408 
1.00350 

10 
9 
8 
7 
6 

•69987 
•  70008 
.  70029 
.  70049 
.  70070 

.71427 
.  71407 
•71386 
•71366 
.71345 
•71325 

•97984 
98041 
.98098 
•98155 
.98213 

35 
34 
33 
32 
31 

55 
56 
57 
58 
59 

-  70608 
-70628 
-  70649 
-70670 
.  70690 

.  70813 
•  70793 
-  70772 
-  70752 
-70731 

.99710 
.99768 
.99826 
.99884 
.99942 

1-00291 
1-00233 
1-00175 
1-00116 
1-00058 

5 

4 
3 
2 
1 

-P 

•70091 

.98270 

1-01761 

30 

6O 

.70711 

•70711 

1  •  00000 

1-00000 

Cos. 

Sin. 

Cot. 

Tan. 

' 

' 

Cos. 

Sin. 

Cot. 

Tan. 

45° 


228 


45° 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
0°  1°  2°  3° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

0 

2 
3 
4 

.00000 
•00000 
•  00000 
.00000 
-00000 

.00000 
.00000 
.00000 
.00000 
.00000 

•00015 
.00016 
.00016 
.00017 
•00017 

.00015 
.00016 
.00016 
.00017 
.00017 

.00061 
.00062 
•00063 
•00064 
•00065 

.00061 
.00062 
.00063 
.00064 
.00065 

•00137 
•00139 
•00140 
•00142 
•00143 

.00137 
.00139 
•00140 
•00142 
•00143 

0 

1 
2 
3 

5 
6 
7 
8 
9 

.00000 
.00000 
.00000 
.00000 
.00000 

.00000 
.00000 
.00000 
.00000 
.  00000 

•00018 
.00018 
.00019 
.00020 
.00020 

.00018 
.00018 
.00019 
.00020 
.00020 

.00066 
.00067 
.00068 
.00069 
.00070 

.00066 
•00067 
.00068 
.00069 
•00070 

•00145 
•00146 
•00148 
•00150 
•00151 

.00145 
.00147 
•00148 
.00150 
•00151 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.00000 
•00001 
.00001 
.00001 
.00001 

.00000 
.00001 
.00001 
.00001 
.00001 

•00021 
•00021 
•00022 
•00023 
•00023 

.00021 
.00021 
.00022 
.00023 
.00023 

•00071 
•00073 
•00074 
•00075 
-00076 

.00072 
.00073 
.00074 
.00075 
.00076 

•00153 
•00154 
•00156 
.00158 
.00159 

.00153 
.00155 
.00156 
.00158 
.00159 

1O 

11 
12 
13 
14 

15 
16 
17 
18 
19 

.00001 
.00001 
.00001 
.00001 

.00002 

.00001 
.00001 
.00001 
.00001 

-00002 

•00024 
•00024 
•00025 
•00026 
-00026 

.00024 
•00024 
.00025 
.00026 

.00026 

•00077 
.00078 
.00079 
.00081 

-00082 

.00077 
.00078 
•00079 
.00081 
.00082 

.00161 
•00162 
•00164 
•00166 
•00168 

.00161 
.00163 
.00164 
•00166 
.00168 

15 
16 
17 
18 

19 

20 

21 
22 
23 
24 

.00002 
-00002 
00002 
-00002 
00002 

.00002 
.00002 
.00002 
.00002 
.00002 

-00027 
•00028 
.00028 
.00029 
-00030 

•00027 
.00028 
.00028 
.00029 
.00030 

•00083 
-00084 
00085 
-00087 
•00088 

.00083 
•00084 
.00085 
•00087 
.00088 

•00169 
•00171 
•00173 
•00174 
•00176 

.00169 
.00171 
•00173 
•00175 
•00176 

20 

21 

22 
23 
24 

25 
26 
27 
28 
29 

-00003 
00003 
-00003 
-00003 
-  00004 

.00003 
.00003 
.00003 
.00003 
•00004 

•00031 
.00031 
•00032 
.00033 
00034 

.00031 
.00031 
.00032 
•00033 
•00034 

-00089 
00090 
•00091 
•00093 
•00094 

•00089 
.00090 
.00091 
•00093 
•00094 

•00178 
•00179 
•00181 
•00183 
•00185 

•00178 
.00180 
.00182 
.00183 
.00185 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

.00004 
.00004 
.  00004 
-  00005 
-00005 

.00004 
.00004 
.00004 
•00005 
.00005 

-00034 
•00035 
•00036 
•00037 
.00037 

•00034 
.00035 
•00036 
.00037 
•00037 

00095 
.00096 
-00098 
.00099 
.00100 

•00095 
-00097 
•00098 
•00099 
•00100 

00187 
00188 
•00190 
•00192 
•00194 

.00187 
•00189 
.00190 
•00192 
.00194 

3O 

31 
32 
33 
34 

35 
36 
37 
38 
39 

00005 
.00005 
.00006 
.00006 
•  00006 

.00005 
.00005 
•00006 
00006 
.00006 

-00038 
.00039 
•00040 
-00041 
-00041 

•00038 
.00039 
•00040 
•  00041 
-00041 

•00102 
•00103 
•00104 
•00106 
-00107 

.00102 
•00103 
•00104 
•00106 
.00107 

•00196 
•00197 
•00199 
•00201 
•00203 

.00196 
.00198 
•00200 
•00201 
-00203 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

-00007 
00007 
-00007 
00008 
-00008 

•00007 
•00007 
.00007 
•00008 
00008 

.  00042 
•00043 
00044 
•00045 
.  00046 

•00042 
•00043 
•00044 
.00045 
•00046 

-00108 
•00110 
•00111 
•00112 
•00114 

•00108 
•00110 
.00111 
•00113 
•00114 

•00205 
•00207 
•00208 
•00210 
00212 

•00205 
•00207 
.00209 
.00211 
.00213 

40 

41 

42 
43 
44 

45 
46 
47 
47 
49 

-00009 
00009 
00009 
-00010 
00010 

•00009 
.00009 
.00009 
.00010 
.00010 

•00047 
00048 
-00048 
•00049 
-00050 

.00047 
.00048 
•00048 
•00049 
.00050 

•00115 
•00117 
•00118 
•00119 
-00121 

.00115 
•00117 
•00118 
•00120 
•00121 

•00214 
•00216 
.00218 
•00220 
•00222 

00215 
.'0216 
.00218 
.00220 
.00222 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

-00011 
00011 
-00011 
00012 
•00012 

.00011 
•00011 
.00011 
•00012 
.00012 

•00051 
•00052 
•00053 
•00054 
•00055 

.00051 
.00052 
.00053 
•00054 
.00055 

•00122 
•00124 
•00125 
•00127 
•00128 

•00122 
•00124 
•00125 
•00127 
•00128 

•00224 
•00226 
•00228 
-00230 
•00232 

.00224 
•00226 
.00228 
.00230 
•00232 

5O 

51 
52 
53 
54 

55 
56 
57 
58 
59 

-00013 
-00013 
-00014 
-00014 
.00015 

.00013 
•00013 
•00014 
.00014 
.00015 

00056 
•00057 
.00058 
•00059 
•00060 

.00056 
•00057 
.00058 
.00059 
00060 

•00130 
•00131 
•00133 
•00134 
-00136 

.00130 
•00131 
00133 
.00134 
.00136 

•00234 
•00236 
-00238 
•00240 
.00242 

•00234 
.00236 
.00238 
.00240 
-00242 

55 
56 
57 
58 
59 

60 

-00015 

•00015 

•00061 

•00061 

•00137 

.00137 

•00244 

.00244 

60 

229 


Inserted  by  permission  from  Webb's  Railroad  Construction. 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
4°  5°  6°  7° 


/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

o 

1 

2 
8 
4 

00244 
•00246 
.00248 
.00250 
•00252 

•00244 
.00246 
.00248 
.00250 
.00252 

.00381 
00383 
•00386 
-00388 
.00391 

.00382 
.00385 
.00387 
.00390 
•00392 

-00548 
•00551 
.00554 
•00557 
-00560 

•00551 
•00554 
•00557 
•00560 
•00563 

.00745 
•00749 
00752 
.00756 
.00760 

•00751 
•00755 
•00758 
•00762 
00765 

0 

1 
2 
3 
4 

5 
6 
7 
8 
9 

.00254 
.00256 
.00258 
•00260 
.00262 

.00254 
.00257 
•00259 
.00261 
00263 

.00393 
•00396 
.00398 
.00401 
.00404 

.00395 
•00397 
•00400 
•  00403 
•00405 

•00563 
.00566 
•00569 
•00572 
.00576 

•00566 
.00569 
.00573 
.00576 
•00579 

00763 
-00767 
•00770 
-00774 
.00778 

•00769 
•00773 
•00776 
•00780 
-00784 

6 
7 
8 
9 

1O 

11 
12 
13 
1.4 

00264 
-00266 
.00269 
.00271 
00273 

•00265 
.00267 
.00269 
.00271 
•00274 

•  00406 
.00409 
•00412 
.00114 
•00417 

•00408 
•00411 
.00413 
•00416 
•00419 

00579 
.00582 
00585 
00588 
•00591 

.00582 
.00585 
•00588 
•00592 
•00595 

•00781 
•00785 
-00789 
00792 
•00796 

•00787 
-00791 
•00795 
•00799 
•00802 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

-00275 
.00277 
.00279 
.00281 
-00284 

-00276 
.00278 
•00280 
•00282 

•00284 

.00420 
.00422 
.00425 
.00428 
-00430 

•00421 
.00424 
.00427 
•00429 
-00432 

00594 
-00598 
-00601 
.00604 
-00607 

•00598 
•00601 
•00604 
.00608 

•00611 

.  00800 
.  00803 
•00807 
•00811 
-00814 

.00806 
.00810 
•00813 
•00817 

.00821 

15 
16 
17 
18 
19 

20 

21 
22 
23 

24 

-00286 
-00288 
•OC290 
.00293 
-00295 

•00287 
•00289 
.00291 
•00293 
•00296 

.00433 
•00436 
•00438 
-  00441 
•00444 

.00435 
•00438 
•'00440 
.00443 
.00446 

-00610 
•00614 
•00617 
•00620 
-00623 

00614 
•00617 
•00621 
-00624 
•00627 

•00818 
•00822 
•00825 
•00829 
-00833 

.00825 
•00828 
•00832 
.00836 
.00840 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

•00297 
•00299 
•C0301 
•00304 
-00306 

.00298 
.00300 
.00302 
.00305 
.00307 

•00447 
.  00449 
•00452 
•00455 
•00458 

.00449 
.00451 
.00454 
.00457 
.00460 

•00626 
•00630 
-00633 
•00636 
-00640 

.00630 
.00634 
.00637 
•00640 
•00644 

•00837 
•00840 
.00844 
•00848 
•00852 

.00844 
•00848 
•00851 
.00855 
.00859 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

•00308 
.00311 
•C0313 
-00315 
-00317 

.00309 
•00312 
.00314 
•00316 
.00318 

•00460 
•00463 
•00466 
•00469 
•00472 

•004b3 
.00465 
.00468 
.00471 
•00474 

•00643 
-00646 
•00649 
-00653 
-00656 

•00647 
.00650 
•00654 
•00657 
•00660 

.00856 
00859 
-00863 
•00867 
•00871 

-00863 
.00867 
-00871 
-00875 
.00878 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

•00320 
•00322 
•00324 
-00327 
-00329 

.00321 
.00323 
.00328 
.00328 
.00330 

.00474 
.00477 
•00480 
-00483 
•00486 

•00477 
.00480 
-00482 
.00485 
-00488 

•00659 
•00663 
-00666 
•00669 
-00673 

.00664 
-00667 
-00671 
-00674 
•00677 

•00875 
-OC878 
-00882 
-00886 
-00890 

-00882 
-00886 
•00890 
-00894 
•00898 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

•00332 
•00334 
-00336 
-00339 
•00341 

.00333 
•00335 
•00337 
•00340 
.00342 

•00489 
•00492 
•00494 
•00497 
•00500 

•00491 
.00494 
-00497 
.00500 
.00503 

-00676 
•00680 
•00683 
-00686 
-00690 

-00681 
.00684 
.00688 
-00691 
•00695 

•00894 
•00898 
•00902 
•00908 
•00909 

•00902 
-00906 
-00910 
-00914 
•00918 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

•00343 
•00346 
•00348 
00351 
00353 

.00345 
•00347 
•00350 
.00352 
•00354 

•00503 
•00506 
•00509 
•00512 
•00515 

-00506 
-00509 
-00512 
•00515 
.00518 

-00693 
00607 
•00700 
•00703 
.00707 

-00698 
-00701 
-00705 
.00708 
.00712 

•00913 
00917 
00921 
00925 
•00929 

-00922 
-00926 
.00930 
-00934 
00938 

45 
46 
47 
4& 
49 

50 

5i 
52 
53 
54 

•00356 
.00358 
00361 
•0036d 
-00365 

•00357 
.00359 
.00362 
•00364 
.00367 

•00518 
•00521 
•00524 
•00527 
.00530 

-00521 
-00524 
-00527 
.00530 
.00533 

•  007.10 
•00714 
•00717 
•00721 
.00724 

-00715 
.00719 
-00722 
-00726 
-00730 

.00933 
00937 
•00941 
.00945 
.00949 

-00942 
-00946 
.00950 
.00954 
•00958 

50 
51 
52 
53 
54 

55 
56 
57 
58 
59. 

•00368 
.00370 
•00373 
.00375 
-00378 

.00369 
.00372 
•00374 
00377 
•00379 

.00533 
.00536 
•00539 
.00542 
00545 

-00536 
-00539 
•00542 
-00545 
00548 

•00728 
-00731 
•00735 
•00738 
•00742 

-00733 
.00737 
-00740 
-00744 
•00747 

•00953 
•00957 
•00961 
.00965 
•00969 

.00962 
-00966 
-00970 
.00975 
-00979 

55 
56 
57 
58 
,59 

60 

•00381 

.00382 

00548 

-00551 

-00745 

.00751 

•00973 

-00983 

60 

230 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
8°  9°  10°  11° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

/ 

0 

2 
3 
4 

.00973 
•00977 
.00981 
.00985 
-00989 

.00983 
.00987 
•00991 
•00995 
.00999 

•01231 
.01236 
•01240 
.01245 
.01249 

.01247 
.01251 
.01256 
.01261 
•01265 

-01519 
.01524 
.01529 
.01534 
.01540 

•01543 
.01548 
.01553 
.01558 
.01564 

-01837 
-01843 
•01843 
.01854 
.01860 

•01872 
.01877 
•01883 
.01889 
•01895 

O 

2 
3 

5 
6 
7 

9 

00994 
00998 
.01002 
.01006 
.01010 

-01004 
•01008 
.01012 
.01016 
.01020 

•01254 
•01259 
•01263 
-01268 
-01272 

•01270 
.01275 
.01279 
-01284 
.01289 

•01545 
•01550 
.01555 
•01560 
.01565 

.01569 
.01574 
.01579 
•01585 
.01590 

•01865 
01871 
•01876 
.01382 
-01888 

.01901 
.01906 
.01912 
.01918 
•01924 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.01014 
•01018 
.01022 
•01027 
.01031 

•01024 
.01029 
.01033 
-01037 
.01041 

•01277 
.01282 
•01286 
.01291 
.01296 

.01294 
•01298 
•01303 
•01308 
•01313 

•01570 
.01575 
•01580 
.01586 
.01591 

.01595 
.01601 
.01606 
.01611 
.01616 

•01893 
•01899 
•01904 
•01910 
.01916 

.01930 
•01936 
.01941 
•01947 
.01953 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

.01035 
-01039 
•01043 
-01047 

.01052 

.01046 
.01050 
-01054 
•01059 
.01063 

.01300 
.01305 
•01310 
•01314 
01319 

•01318 
•01322 
•01327 
•01332 
•01337 

•01596 
-01601 
.01606 
•01612 
-01617 

•01622 
.01627 
.01633 
.01638 
.01643 

•01921 
.01927 
.01933 
•01939 
•01944 

•01959 
•01965 
•01971 
.01977 
•01983 

15 
16 
17 
18 
-19 

20 

21 
22 
23- 
24 

-01056 
01060 
01064 
-01069 
•01073 

.01067 
.01071 
.01076 
.01080 
.01084 

•01324 
•01329 
-01333 
•01338 
.01343 

.01342 
•01346 
•01351^ 
.01356 
.01361 

.01622 
•01627 
.01632 
•01638 
•01643 

•01649 
•01654 
.01659 
.01665 
.01670 

-01950 
•01956 
•01961 
•01967 
•01973 

.01989 
•01995 
•02001 
•02007 
•02013 

2O 

21 
22 
23 
24 

25 
26 
27 
28 
29 

.01077 
.01081 
.01086 
-01090 
.01094 

.01089 
.01093 
.01097 
.01102 
.01106 

.01348 
•01352 
.01357 
•01362 
.01367 

.01366 
.01371 
.01376 
.01381 
.01386 

•01648 
-01653 
•01659 
•01664 
•01669 

.01676 
.01681 
.01687 
.01692 
.01698 

-01979 
•01984 
•01990 
-01996 
•02002 

•02019 
•02025 
.02031 
.02037 
.02043 

25 
28 
27 
28 

29 

30 

31 
32 
33 
34 

•01098 
.01103 
-01107 
.01111 
.01116 

.01111 
.01115 
.01119 
.01124 
.01128 

•01371 
.01376 
•01381 
.01386 
.01391 

.01391 
•01395 
•01400 
.01405 
.01410 

.01675 
•01680 
•01685 
•01690 
.01696 

.01703 
.01709 
.01714 
.01720 
.01725 

.02008 
.02013 
•02019 

.02025 
-02031 

.02049 
.02055 
.02061 
•02067 
.02073 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

•01120 
.01124 
.01129 
•01133 
-01137 

.01133 
.01137 
.01142 
.01146 

-01151 

•01396 
-01400 
.01405 
•01410 

•01415 

.01415 
.01420 
•01425 
.01430 

•  OUS'i 

•01701 
•01706 
•01712 
-01717 

-01723 

.01731 
-01736 
.01742 
.01747 
.01753 

•02037 
.02042 
•02048 
•02054 
.02060 

.02079 
.02085 
.02091 
•02097 
•02103 

35 
36 
37 
38 
39 

40 

41 
42 
43 

44 

01142 
.01148 
•01151 
-01155 
-01159 

•01155 
.01160 
•01164 
.01169 
01173 

-01420 
.01425 
•01430 
-01435 
.01439 

•01440 
.01445 
01450 
-01455 
.01461 

•01728 
01733 
.01739 
.01744 
.01750 

•01758 
.01764 
.01769 
.01775 
.01781 

.02066 
.02072 
•02078 
•02084 
-02090 

•02110 
•02116 
•02122 
.02128 
•02134 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

.01164 
-01168 
01173 
-01177 
-01182 

.01178 
-01182 
.01187 
.01191 
•01196 

•01444 
•01449 
•01454 
•01459 
.01464 

.01466 
•01471 
.01476 
•01481 
•01486 

.01755 
•01760 
01766 
•01771 
.01777 

01786 
.01792 
.01793 
.01803 
.01809 

•02095 
•02101 
•02107 
.02113 
-02119 

•02140 
•02146 
•02153 
•02159 
•02165 

45 
48 
47 
48 
49 

50 

51 
R2 
53 
54 

01186 
-01191 
.01195 
-01200 
-01204 

.01200 
•01205 
.01209 
.01214 
.01219 

•01469 
•01474 
•01479 
•01484 
.01489 

•01491 
•01496 
•01501 
•01506 
.01512 

•01782 
•01788 
.01793 
.0179^ 
-01804 

.01815 
.01820 
•01826 
•01832 
01837 

.02125 
•02131 
.02137 
.02143 
•02149 

.02171 
.02178 
•02184 
•02190 
•02196 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59, 

.01209 
.01213 
-01218 
-01222 
-01227 

.01223 
•01228 
.01233 
.01237 
.01242 

•01494 
•01499 
•01504 
•01509 
-01514 

.01517 
.01522 
.01527 
.01532 
.01537 

.01810 
•01815 
•01821 
.01826 
.01832 

•01843 
•01849 
.01854 
•01860 
•01866 

.02155 
•02161 
.02167 
•02173 
.02179 

•02203 
•02209 
•02215 
-02221 
.02228 

55 
58 
57 
58 
59 

6O 

01231 

.01247 

.01519 

.01543 

.01837 

.01872 

•02185 

.02234 

60 

231 


TABLE  11— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
12°  13°  14°  15° 


t 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

o 

1 

2 
3 
4 

02185 
02191 
02197 
.02203 
02210 

.02234 
.02240 
.02247 
.02253 
.02259 

.02563 
.02570 
.02576 
•02583 
.02589 

•02630 
.02637 
•02644 
.02651 
•02658 

02970 
02977 
02985 
02992 
02999 

.03081 
•03069 
.03076 
•03084 
•03091 

03407 
•03415 
-03422 
•03430 
•03438 

•03528 
•03536 
•03544 
•03552 
•03560 

O 

2 
3 

4 

5 
6 
7 
8 
9 

.02216 
.02222 
.02228 
02234 
02240 

.02266 
.02272 
.02279 
.02285 
-02291 

•02596 
•02602 
.02609 
-02616 
.02622 

•02665 
.02672 
.02679 
.02686 
.02693 

03006 
03013 
03020 
•03027 
•03034 

•03099 
.03106 
•03114 
.03121 
.03129 

-03445 
•03453 
03460 
•03468 
.03476 

.03568 
.03576 
-03584 
-03592 
•03601 

5 
6 
7 

8 

9 

10 

11 
12 
13 

14 

.02246 
•02252 
•02258 
.02265 
02271 

.02298 
-02304 
.02311 
.02317 
•02323 

•02629 
-02635 
•02642 
.02649 
.02655 

.02700 
.02707 
•02714 
.02721 
.02728 

.03041 
•03048 
•03055 
•03063 
•03070 

•03137 
.03144 
.03152 
.03159 
•03167 

•03483 
•03491 
-03498 
-03506 
-03514 

-03609 
-03617 
-03625 
-03633 
-03642 

10 

11 
12 
13 
14 

15 
16 
17 
18 

19 

02277 
.02283 
•02289 
.02295 
•02302 

•02330 
.02336 
.02343 
.02349 
.02356 

.02662 
•02669 
•02675 
•02682 
.02689 

.02735 
.02742 
.02749 
.02756 

•02763 

.03077 
.03084 
.03091 
.03098 
•03106 

•03175 
.03182 
•03190 
-03198 
-03205 

•03521 
•03529 
•03537 
•03544 
•03552 

-03650 
-03658 
•03666 
.03674 
•03683 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

•02308 
•02314 
•02320 
•02327 
.  02333 

.02362 
•02369 
•02375 
.02382 
•02388 

•02696 
•02702 
•02709 
.02716 
•02722 

.02770 
.02777 
.02784 
.02791 
.02799 

•03113 
•03120 
•03127 
•031S4 
•03142 

•C3213 
•03221 
•03228 
.03236 
.03244 

.03560 
•03567 
•03575 
.03583 
•03590 

-03691 
-03699 
.03708 
•03716 
•03724 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

.02339 
•02345 
•02352 
-02358 
•02364 

.02395 
•02402 
•02408 
•02415 
.02421 

•02729 
•02736 
•02743 
.02749 
.02756 

.02806 
.02813 
.02820 
.02827 
.02834 

.03149 
.03156 
•03163 
-03171 
•03178 

.03251 
.03259 
.03267 
.03275 
•03282 

.03598 
•03606 
.03614 
•03621 
•03629 

•03732 
•03741 
•03749 
•03758 
.03766 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

•02370 
•02377 
-02383 
•02389 
-02396 

•02428 
.02435 
.02441 
.02448 
•02454 

.02763 
•02770 
•02777 
•02783 
•02790 

.02842 
•02C49 
.02856 
.02863 
.02870 

-03185 
•03193 
•03200 
•03207 
•03214 

•03290 
•03298 
•03306 
•03313 
•03321 

•03637 
•03645 
•03653 
•03660 
•03668 

•03774 
•03783 
•03791 
•03799 
•03808 

30 

31 
32 
33 
34 

3o 
36 
37 
38 
39 

•02402 
-02408 
•02415 
02421 

-02427 

.02461 
•02468 
•02474 
.02481 
.02488 

•02797 
•02804 
•02811 
•02818 
-02824 

.02878 
.02885 
.02892 
.02899 

•02907 

•03222 
•03229 
•03236 
•03244 
•03251 

.03329 
-03337 
.03345 
•03353 

03360 

•03676 
•03684 
.03692 
•03699 
•03707 

•03816 
•03825 
-03833 
-03842 
•03850 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

-02434 
-02440 
•02447 
•02453 
•02459 

•02494 
.02501 
•02508 
•02515 
•02521 

•02831 
•02838 
•02845 
02852 
•02859 

.02914 
.02921 
.02928 
.02936 
•02943 

•03258 
•03266 
•03273 
•03281 
•03288 

-03368 
•03376 
•03384 
•03392 
.03400 

03715 
-03723 
•03731 
•03739 
•03747 

•03858 
.03867 
-03875 
.03884 
.03892 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 

•  02466 
•02472 
•02479 
•02485 
•02492 

•02528 
•02535 
•02542 
•02548 
•02555 

•02866 
•02873 
•02880 
•02887 
•02894 

.02950 
.02958 
.02965 
.02972 
.02980 

•03295 
•03303 
•03310 
•03318 
03325 

•03408 
•03416 
.03424 
.03432 
•03439 

•03754 
•03762 
03770 
•03778 
•03786 

.03901 
.03909 
.03918 
.03927 
.03935 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•02498 
-02504 
•02511 
•02517 
-02524 

•02562 
•02569 
•02576 
•02582 
•02589 

•02900 
•02907 
•02914 
•02921 
.02928 

.02987 
•02994 
•03002 
•03009 
.03017 

03333 
•03340 
-03347 
.03355 
•03362 

•03447 
•03455 
.03463 
•03471 
•03479 

•03794 
.03802 
.03810 
•03818 
•03826 

.03944 
-03952 
.03961 
.03969 
•03978 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

-02530 
•02537 
•02543 
•02550 

-02556 

•02596 
•02603 
•02610 
•02617 

-02624 

•09935 
-02942 
02949 
•02956 
•02963 

.03024 
•03032 
.03039 
03046 
•03054 

.03370 
.03377 
03385 
•03392 
.03400 

•03487 
-03495 
.03503 
•03512 
•03520 

.03834 
.03842 
•03850 
•03858 
•03866 

.03987 
-03995 
•04004 
.04013 

.04021 

55 
56 
57 
58 
59 

60 

•02563 

02630 

•02970 

.03061 

03407 

.03528 

•03874 

.04030 

60 

232 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
16°  17°  18°  19° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec 

I 

o 

1 

2 
3 

03874 
03882 
03890 
03898 
03906 

.04030 
.04039 
.04047 
.04056 
.04065 

04370 
04378 
04387 
04395 
04404 

•04569 
•04578 
-04588 
.04597 
-04606 

04894 
04903 
04912 
04921 
04930 

.05146 
-05156 
.05166 
•05176 
.05186 

05448 
05458 
05467 
05477 
05486 

.05762 
-05773 
•05783 
•05794 
.05805 

0 

1 
2 
3 
4 

5 
6 
7 
8 
9 

03914 
J3922 
03930 
03938 
03946 

•04073 
.04082 
.04091 
.04100 
-04108 

04412 
04421 
04429 
04438 
04446 

.04616 
•04625 
04635 
.04644 
.04653 

04939 
04948 
04957 
04967 
04976 

.05196 
•05206 
.05216 
.05226 
•  05'236 

05496 
05505 
05515 
05524 
05534 

•05815 
•05826 
-05836 
•05847 
.05858 

5 
6 
7 
8 
9 

10 

11 

12 
13 
14 

03954 
03963 
03971 
03979 
03987 

.04117 
04126 
.04135 
•04144 
.04152 

04455 
04464 
04472 
04481 
04489 

.04663 
.04672 
•04682 
.04691 
.04700 

04985 
04904 
05003 
05012 
05021 

.05246 
.05256 
.05266 
-05276 
.05286 

.05543 
•05553 
•05562 
•05572 
.05582 

•05869 
-05879 
•05890 
•05901 
.05911 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

03995 
04003 
04011 
04019 

04028 

.04161 
•04170 
•04179 
•04188 
-04197 

04498 
04507 
04515 
04524 
04533 

.04710 
.04719 
.04729 
.04738 

-04748 

05030 
05039 
05048 
05057 
05067 

.05297 
.05307 
.05317 
.05327 
.05337 

•05591 
.05601 
05610 
•05620 
.05630 

.05922 
.05933 
.05944 
.05955 
.05965 

15 
16 
17 
18 
19 

20 

21 

22 
23 
24 

04036 
04044 
04052 
04060 
04069 

•04206 
.04214 
.04223 
.04232 
.04241 

04541 
04550 
0455S 
04567 
04576 

.04757 
.04767 
.04776 
.04786 
.04795 

05076 
05085 
.05094 
.05103 
.05112 

-05347 
.05357 
.05367 
.05378 
-05388 

.05639 
.05649 
.05688 
.05668 
.05678 

•05976 
.05987 
.05998 
.06009 
.06020 

20 

21 
22 
23 
24 

25 
26 
27 
28 

29 

04077 
04085 
04093 
04102 
04110 

•04250 
•04259 
.04268 
.04277 
•04286 

04585 
04593 
.04602 
.04611 
.04620 

.04805 
.04815 
.04824 
04834 
•04843 

.05122 
.05131 
.05140 
•05149 
.05158 

05398 
-05408 
.05418 
.05429 
-05439 

.05687 
•05697 
05707 
•05716 
.05726 

.06030 
.06041 
.06052 
.06063 
.06074 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

04118 
04126 
04135 
04143 
.04151 

•04295 
.04304 
.04313 
.04322 
.04331 

-04628 
•04637 
-04646 
04655 
.04663 

.04853 
.04863 
•04872 
.04882 
.04891 

.05168 
.05177 
.05186 
.05195 
.05205 

.05449 
.05460 
.05470 
.05480 
.05490 

.05736 
.05746 
.05755 
.05765 
.05775 

.06085 
.06096 
.06107 
-06118 
.06129 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

•04159 
.04168 
.04176 
.04184 
.04193 

•04340 
•04349 
•04358 
•04367 
.04376 

.04672 
•04681 
.04690 
-04699 
-04707 

.04901 
•04911 
04920 
•04930 
-04940 

-05214 
.05223 
.05232 
.05242 
.05251 

.05501 
•05511 
.05521 
•05532 

.05542 

.05785 
.05794 
.05804 
.05814 

.05824 

•06140 
.06151 
•06162 
•06173 
.06184 

35 
-  36 
37 
38 

-39 

40 

41 
42 
43 
44 

-04201 
.04209 
.04218 
.04226 
.04234 

.04385 
.04394 
.04403 
.04413 
.04422 

-04716 
.04725 
.04734 
.04743 
-04752 

-04950 
•04959 
•04969 
-04979 
.04989 

.05260 
.05270 
.05279 
.05288 
.05298 

.05552 
.05563 
.05573 
.05584 
.05594 

.05833 
.05843 
.05853 
.05863 
.05873 

.06195 
.06206 
•06217 
-06228 
.06239 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

.04243 
.04251 
04260 
•04268 
•04276 

.04431 
•04440 
.  04449 
•04458 
.04468 

-04760 
•04769 
.04778 
.04787 
.04796 

.04998 
•05008 
.05018 
.05028 
.05038 

.05307 
.05316 
.05326 
.05335 
.05344 

.05604 
.05615 
.05625 
.05636 
.05646 

.05882 
.05892 
-05902 
.05912 
.05922 

.06250 
.06261 
-06272 
.06283 
.06295 

45 
46 
47 
48 
49 

50 

51 

52 
53 
54 

04285 
.04293 
•04302 
.04310 
-04319 

•04477 
•04486 
.04495 
.04504 
.04514 

•04805 
.04814 
-04323 
.04832 
-04841 

•05047 
.05057 
.05067 
•05077 
•05087 

•05354 
.05363 
.05373 
.05382 
-05391 

.05657 
.05667 
.05678 
.05688 
-05699 

.05932 
.05942 
.05951 
.05961 
.05971 

.06306 
.06317 
.06328 
.06339 
.06350 

5O 

51 
52 
53 
54 

55 
56 
57 
58 
59 

.04327 
.04336 
•04344 
.04353 
.043fil 

•04523 
•04532 
.04541 
.04551 
-04560 

-04850 
•04858 
-04867 
-04876 
04885 

•05097 
•05107 

.05116 
.05126 
.05136 

.05401 
.05410 
.05420 
.05429 
.05439 

.05709 
.05720 
.05730 
.05741 
.05751 

.05981 
-05991 
-06001 
.06011 
.06021 

.06362 
.06373 
.06384 
.06395 
-06407 

55 
56 
57 
58 
59 

60 

•04370 

-04569 

-04894 

•05146 

.05448 

.05762 

.06031 

.06418 

60 

2 

33 

TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
30°  21°  22°  23° 


t 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

f 

0 

2 
3 
4 

-06031 
.06041 
•06051 
•06061 
.06071 

•06418 
.06429 
.06440 
.06452 
06463 

•06642 
•06652 
•06663 
.06673 
•06684 

.07115 
.07126 
.07138 
.07150 
.07162 

.07282 
.07293 
.07303 
.07314 
.07325 

.07853 
•07866 
.07879 
•07892 
•07904 

•07950 
•07961 
•07972 
•07984 
-07995 

.08636 
•08649 
•08663 
.08676 
.08690 

O 

2 
3 

4 

5 
6 
7 
8 
9 

•06081 
•06091 
.06101 
•06111 
06121 

.06474 
.06486 
06497 
.06508 
.06520 

•06694 
•06705 
•06715 
.06726 
•06736 

.07174 
.07186 
•07199 
.07211 
.07223 

•07336 
•07347 
-07358 
•07369 
-07380 

.07917 
07930 
•07943 
•07955 
.07968 

-C8006 
-C8018 
-08029 
•C8041 
.08052 

.08703 
•08717 
•08730 
•08744 
.08757 

5 
6 
7 
8 

9 

10 

11 
12 
13 
14 

•06131 
•06141 
•06151 
.06161 
.06171 

.06531 
.06542 
.06554 
.06565 
•06577 

.06747 
•06757 
•06768 
•06778 
.06789 

.07235 
•07247 
.07259 
.07271 
.07283 

•07391 
•07402 
•07413 
•07424 
.07485 

.07981 
.07994 
.08006 
.08019 
•08032 

•08064 
•08075 
•08086 
•08098 
•08109 

•08771 
•08784 
.08798 
.08811 
.08825 

10 

11 
12 
IS 
14 

15 
16 
17 
18 
19 

•06181 
•06191 
•06201 
•06211 
.06221 

•06588 
.06600 
.06611 
.06622 

.06634 

•06799 
•06810 
•06820 
•06831 
.06841 

•07295 
.07307 
.07320 
.07332 
•07344 

.07446 
.07457 
•07468 
.07479 
.07490 

.08045 
.08058 
.08071 
-08084 
.08097 

.08321 
.08132 
.08144 
.08155 
08167 

.08839 
.08852 
.08866 
.08880 
.08893 

15 
13 
17 
18 

19 

2O 

21 
22 
23 
24 

•06231 
•06241 
•06252 
•06262 
.06272 

.06645 
.06657 
•06668 
.06680 
.06691 

•06852 
.06863 
•06873 
•06884 
•06894 

-07356 
.07368 
.07380 
.07393 
.07405 

.07501 
.07512 
.07523 
.07534 
.07545 

•08109 
.08122 
.08135 
.08148 
.08161 

.08178 
•08190 
•08201 
•08213 
•08225 

.08907 
.08921 
08934 
.08948 
.08962 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

-06282 
.06292 
06302 
06312 
06323 

.06703 
.06715 
.06726 
.06738 
.06749 

•06905 
.06916 
.06926 
.06937 
.06948 

.07417 
.07429 
.07442 
.07454 
.07486 

.07556 
.07568 
.07579 
.07590 
•07601 

.08174 
.08087 
.08200 
.08213 
•08226 

•08236 
08248 
•08259 
•08271 
08282 

.08975 
.08989 
.09003 
•09017 
.09030 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

08333 
06343 
06353 
06363 
06374 

.06761 
.06773 
.06784 
.06796 
.06807 

.06958 
.06969 
•06980 
.06990 
.07001 

.07479 
.07491 
.07503 
.07516 
.07528 

.07612 
•07623 
.07634 
.07645 
•07657 

.08239 
.08252 
•08265 
.08278 
.08291 

•08294 
•08306 
•08317 
.'08329 
.  08340 

.09044 
.09058 
.09072 
.09086 
.09099 

30 

31 
32 
.33 
34 

35 
36 
37 
38 
39 

06384 
06394 
06404 
06415 
06425 

.06819 
.06831 
•06843 
•06854 
•06866 

.07012 
.07022 
.07033 
07044 

07055 

.07540 
.07553 
.07565 
.07578 
•07590 

•07668 
.07679 
.07690 
-07701 
•07713 

•08305 
.08318 
.08331 
•08344 
.08357 

•08352 
•08364 
-08375 
•08387 
•08399 

•09113 
•09127 
.09141 
.09155 
.09169 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

06435 
06445 
06456 
06466 
06476 

.06878 
.06889 
•06901 
.06913 
.06925 

07065 
07076 
07087 
07098 
07108 

.07J02 
.07615 
.07627 
•07U40 
.07652 

.07724 
.07735 
•07746 
.07757 
07769 

.08370 
.08383 
08397 
.08410 
.08423 

•08410 
•08422 
08434 
•08445 
•08457 

•09183 
.09197 
.09211 
•09224 
.09238 

40 

41 
42 
48 
44- 

45 
46 
47 
48 
49 

06486 
06497 
06507 
06517 
06528 

.06936 
-06948 
.06960 
.06972 
06984 

07119 
07130 
07141 
07151 
07162 

.07665 
.07677 
.07690 
.07702 
.07715 

07780 
07791 
07802 
07814 
07825 

.08436 
.08449 
.08463 
•08476 
.08489 

08469 
•08481 
.08492 
.08504 
.08516 

.09252 
.09266 
•09280 
•09294 
09308 

45 
46 
47 
48 
49 

5O 

51 
52 
53 
54 

06538 
06548 
06559 
06569 
06580 

.06995 
.07007 
.07019 
-07031 
•  07043 

07173 
07184 
07195 
07206 
07216 

.07727 
.07740 
.07752 
.07765 
.07778 

07836 
07848 
07859 
07870 
07881 

.08503 
.08516 
.08529 
.08542 
.08556 

08528 
08539 
08551 
08563 
08575 

.09323 
-09337 
.09351 
•09365 
09379 

50 

51 

52 
53 
54 

55 
56 
57 
58 
59 

06590 
06600 
06611 
06621 
06632 

.07055 
.07067 
.07079 
.07091 

.07103 

07227 
07238 
07249 
07260 
07271 

.07790 
•07803 
.07816 
•07828 
-07841 

07893 
07904 
07915 
07927 
07938 

.08569 
•08582 
.08596 
.08069 
•08623 

08586 
08598 
08610 
08622 
08634 

•09393 
•09407 
•09421 
•09435 
09449 

55 
56 
57 
58 

59 

60 

06642 

.07115 

07282 

•07853 

07950 

.08636 

08645 

•09464 

60 

234 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
24°  25°  26°  27° 


0 

2 
3 

~5~ 
6 
7 
8 
_9_ 
10 
11 
12 
13 
14 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

t 

~ 

2 
3 
4 
5 
6 
7 
8 
9 

08645 
.08657 
•08669 
•08681 
.08693 

.09464 
.09478 
.09492 
.09506 
.09520 

.09369 
.09382 
.09394 
•09406 
-09418 

•10338 
.10353 
.10368 
.10383 
•10398 

10121 
•10133 
.10146 
.10159 
-10172 

•11260 
•11276 
•11292 
•11308 
•11323 

•10899 
•10913 
•10926 
•10939 
-10952 

•  12233 
-1224b 
•12266 
-12283 
•12299 

.08705 
.08717 
.08728 
.08740 
.08752 

•09535 
•09549 
.09563 
.09577 
.09592 

•09431 
•09443 
•09455 
09468 
.09480 

-10413 
.  10428 
.10443 
.10458 
.10473 

•10184 
•10197 
.10210 
•10223 
•10236 

•11339 
11355 
•11371 
•11387 
•11403 

•10965 
•10979 
-10992 
-11005 
.11019 

-12316 
-12333 
-  12349 
.12366 
-12383 

.08764 
•08776 
.08788 
.088CO 
.08812 

.09608 
•09620 
.09635 
.09649 
.09663 

.09493 
.09505 
•09517 
•09530 
.09542 

.10488 
•10503 
10518 
•10533 
.10549 

.10248 
•10261 
•10274 
.10287 
.10300 

•11419 
•11435 
.11451 
-11467 
.11483 

•11032 

•11045 
.11058 
11072 
.11085 

•12400 
•12416 
•12433 
•12450 
.12467 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

.08824 
•08836 
.08848 
.08860 
.08872 

.09678 
.09692 
.09707 
.09721 
-09735 

09554 
•09567 
•09579 
•09592 
.09604 

.10564 
.10579 
.10594 
•10609 
.10625 

•10313 
•10326 
.10338 
•10351 
•10364 

•11499 
11515 
•11531 
•11547 
.11563 

•11098 
•11112 
•11125 
-11138 
•11152 

.12484 
.12501 
.12518 
-125b4 
-12551 

15 
16 
17 

18 
19 

20 

21 
22 
23 
24 

.08884 
•08896 
•08908 
.08920 
.08932 

.09750 
.09764 
.09779 
.09793 
.09808 

-09617 
•09629 
•09642 
•09654 
.09666 

•10640 
•10655 
•10670 
.10686 
-10701 

•10377 
•10390 
•  10403 
•10416 
10429 

•11579 
•11595 
-11611 
•11627 
-11643 

•11165 
•11178 
•11192 
•11205 
•11218 

•12568 
-12585 
.12602 
.12619 
12636 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

25 
26 
27 
28 
29_ 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

.08944 
.08956 
.08968 
.08980 
.08992 

.09822 
•09837 
.09851 
.09866 
•09880 

•09679 
•09691 
.09704 
•09716 
•09729 

.10716 
.10731 
•10747 
•10762 
•10777 

.  10442 
•10455 
.10468 
•10481 
.10494 

•11659 
-11675 
•11691 
-11708 
-11724 

•11232 
•11245 
•11259 
•11272 
.11285 

-12653 
-12670 
-12687 
.12704 
•12721 

.09004 
.09016 
.09028 
-.09040 
.09052 

.09895 
•09909 
.09924 
-09939 
•09953 

•09741 
.09754 
•09767 
.09779 
.09792 

.10793 
10808 
10824 
-10839 
-10854 

•10507 
•10520 
.10533 
•10546 
.10559 

.11740 
•11756 
-11772 
-11789 
11805 

•11299 
•11312 
•11326 
•11339 
.11353 

.12738 
•12755 
•12772 
•12789 
.  12807 

.09064 
.09076 
.09089 
.09101 

.09113 

.09968 
•09982 
.09997 
.10012 
.10026 

.09804 
09817 
09829 
•09842 
.09854 

.10870 
.10885 
-10901 
•10916 
.10932 

•10572 
-10585 
.10598 
•10611 
•  10624 

•11821 
.11838 
•11854 
•11870 
-11886 

•11366 
•11380 
•11393 
•11407 
•11420 

•12824 
•12841 
-12858 
.12875 
-12892 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

•09125 
•09137 
.09149 
.09161 
.09174 

.10041 
.10055 
.10071 
.10085 
.10100 

•09867 
•09880 
•09892 
•09905, 
.09918 

.  10947 
•10963 
.10978 
.10994 
•11009 

•10637 
.10650 
.10663 
•10676 
.10689 

•11903 
•11919 
•11936 
•11952 
.11968 

•11434 

•11447 
•11461 
•11474 
.11488 

12910 
-12927 
•12944 
•12961 
•12979 

4O 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

.09186 
•09198 
•09210 
.09222 
-09234 

.10115 
.10130 
.10144 
.10159 
.10174 

•09930 
•09943 
•09955 
•09968 
-09981 

-11025 
•11041 
-11056 
•11072 
•11087 

•10702 
•10715 
-10728 
•10741 
•10755 

•11985 
•12001 
•12018 
•12034 
•12051 

•11501 
•11515 
-11528 
-11542 
-11555 

•12996 
•13013 
-13031 
•  13048 
13065 

.09247 
•09259 
•09271 
-09283 
.09296 

•10189 
.10204 
.10218 
.10233 
.10248 

•09993 
•10006 
•10019 
•10032 
.  10044 

•11103 
-11119 
•11134 
.11150 
•11166 

•10768 
•10781 
10794 
-10807 
10820 

•12067 
•  12084 
•12100 
•12117 
•12133 

-11569 
•11583 
-11596 
11610 
-11623 

.  13083 
•13100 
•13117 
•13135 
13152 

55 
56 
57 
58 
59 

•09308 
•09320 
.09332 
-09345 
.09357 

.10263 
•10278 
.10293 
.10308 
.10323 

•10057 
•10070 
•10082 
•10095 
•10108 

.11181 
.11197 
.11213 
.11229 

•11244 

.10833 
•10847 
•10860 
•10873 
•10886 

-12150 
•12166 
•12183 
.12199 
•12216 

•11637 
•11651 
•11664 
•11678 

.11692 

•13170 
•13187 
.13205 
•13222 
.13240 

55 
56 
57 
58 
59 

60 

.09369 

.10338 

•10121 

.11260 

.10899 

.12233 

•11705 

•13257 

60 

235 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS, 
28°  29°  30°  31° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

0 

2 
3 
4 

•11705 
•11719 
.11733 
•11746 
•11760 

.13257 
.13275 
•13292 
•13310 
•13327 

.12538 

.12552 
•12566 
•12580 
•12595 

•14335 
-14354 
.14372 
.14391 
•  14409 

•13397 
•13412 
•13427 
•13441 
-13456 

•15470 
.15489 
•15509 
.15528 
•15548 

•14283 
.14298 
•14313 
•14328 
.  14343 

•16663 
•16684 
•16704 
.16725 
.16745 

O 

1 
2 
3 

•4 

5 
6 
7 
8 
9 

•11774 
•11787 
-11801 
•11815 
•11828 

.13345 
.13362 
•13380 
.13398 
•13415 

•12609 
•12623 
•12637 
•12651 
•12665 

.14428 
-14446 
•14465 
.14483 
•14502 

•13470 
•13485 
-13499 
-13514 
-13529 

.15567 
•15587 
•15606 
-15626 
-15645 

•14358 
•14373 
•14388 
•  14403 
-14418 

.16766 
.16786 
.16806 
•16827 
•16848 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

•11842 
•11856 
•11870 
•11883 
•11897 

•13433 
•13451 
•13468 
•13486 
-13504 

•12679 
•12694 
.12708 
•12722 
•12736 

•14521 
.14539 
.14558 
.14576 
•14595 

.13543 
•13558 
.13573 
-13587 
-13602 

-15665 
-15684 
•15704 
•15724 
•15743 

.  14433 
•  14449 
.  14464 
•  14479 
.  14494 

•16868 
•16889 
-16909 
.16930 
-16950 

10 

11 

12 
13 
14 

15 
16 
17 
18 
19 

.11911 
•11925 
•11938 
•11952 
.11966 

•13521 
•13539 
•13557 
.13575 
•13593 

•12750 
•12765 
.12779 
.12793 
•12807 

•14614 
.14632 
.14651 
•14670 
-14689 

-13616 
-13631 
-13646 
-13660 
•13675 

•15763 
•15782 
.15802 
•15822 
-15841 

•14509 
•14524 
•14539 
•14554 
.14569 

-16971 
•16992 
•17012 
-17033 
-17054 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

.11980 
.11994 
•12007 
•12021 
.12035 

•13610 
.13628 
.13646 
•13664 
•13682 

.12822 
•12836 
•12850 
•12864 
•12879 

-14707 
.14726 
.14745 
•14764 
.14782 

-13690 
-13705 
-13719 
-13734 
-  13749 

.15861 
.15881 
.35901 
.15920 
.15940 

-14584 
-14599 
-14615 
•14630 
•14645 

•17075 
•17095 
•17116 
-17137 
•17158 

20 

21 

22 
23 

24 

25 
26 
27 
28 
29 

.  12049 
•12063 
-12077 
•12091 
12104 

-13700 
•13718 
.13735 
.13753 
.13771 

.12893 
•12907 
•12921 
.12936 
.12950 

.14801 
.14820 
.14839 
.14858 
.14877 

•13763 
•13778 
•13793 
•13808 
.13822 

•15960 
•15980 
.16000 
•16019 
-16039 

•14660 
•14675 
•14690 
•14706 
.14721 

-17178 
.17199 
.17220 
•17241 
•17262 

25 
28 
27 
28 
29 

30 

31 
32 
33 
34 

•12118 
.12132 
•12146 
12160 
12174 

.13789 
.13807 
.13825 
.13843 
.13861 

•12964 
.12979 
.12993 
.13007 
.13022 

.14896 

.14914 
.14933 
-14952 
.14971 

•33837 
•13852 
•13867 
•13881 
-13896 

•16059 
.16079 
.16099 
.16119 
.16139 

•14736 
•14751 
•14766 
•14782 
•14797 

•17283 
•17304 
•17325 
•17346 
•17367 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

.12188 
•12202 
•12216 
•12230 
•12244 

•13879 
.13897 
.13916 
•13934 

13952 

.13036 
.13051 
•13065 
•13079 
.13094 

.14990 
.15009 
•15028 
.15047 
.15066 

-13911 
-13926 
-13941 
-13955 
-18970 

.16159 
.16179 
•16199 
•16219 
-16239 

-14812 
.14827 
•  14843 
•14858 
-14873 

•17388 
•17409 
•37430 
•17451 
-17472 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.12257 
•12271 
•12285 
.12299 
.12313 

•13970 
.13988 
•14006 
•14024 
•  14042 

.13108 
•13122 
•13137 
•13151 
.13166 

•15085 
-1510u 
.15124 
-15143 
•15162 

13985 
-  14000 
-14015 
-  14030 
-14044 

.16259 
.16279 
.16299 
9.  16319 
-16339 

•14888 
•14904 
•34819 
•14934 
•34949 

•17493 
•17514 
-17535 
•17556 
•17577 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

-12327 
.32341 
.12355 
.12369 
12383 

•14061 
•  14079 
•14097 
•14115 
•14134 

•13180 
-13195 
.13209 
.13223 
.13238 

-15181 
.15200 
.15219 
.15239 
.15258 

-14059 
14074 
-14089 
•14104 
14119 

16359 
•16380 
.16400 
•16420 
•16440 

•14965 
•  3  4980 
•14995 
•15011 
15026 

-17598 
-17620 
•17641 
-17662 
•176S3 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

.1239'/ 
•12411 
.12425 
.  12439 
.12454 

.14152 
.14170 
•14188 
.14207 
•14225 

.13252 
.13267 
.13281 
•13296 
•13310 

.15277 
.15296 
.15315 
.15335 
.15354 

•14134 
14149 
•14164 
14179 
•14194 

.16460 
16481 
•16501 
.16521 
.16541 

15041 
15057 
15072 
15087 
35103 

.17704 
•17726 
•17747 
.17768 
.17790 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

.12468 
•12482 
•12496 
•12510 
.12524 

.14243 
14262 
•14280 
.14299 
.14317 

.13325 
.13339 
.13354 
.13368 
.13383 

.15373 
.15393 
.15412 
.15431 
-15451 

14208 
•14223 
•14238 
•14253 
14268 

•16562 
•16582 
•16602 
•16623 
-16643 

15118 
15134 
15149 
15164 
15180 

•17811 
-17832 
•17854 
•17875 
.17896 

55 
56 
57 
58 
59 

60 

.12538 

.14335 

•13397 

.15470 

-14283 

•16663 

15195 

•17918 

60 

236 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
32°  33°  34°  35° 


/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

t 

~ 

2 
3 

a 

6 
7 

8 

9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

o 

2 
3 

~5~ 
7 

9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19- 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

•15195 
.15211 
.15226 
.15241 
•15257 

•17918 
•17939 
.17961 
.17982 
.18004 

•16133 
.16149 
.16165 
-16181 
-16196 

.19236 
•19259 
•19281 
.19304 
-19327 

.17096 
.17113 
•17129 
.17145 
.17161 

.20622 
.20645 
•20669 
•20693 
-20717 

-18085 
-18101 
-18118 
.18135 
-18152 

-22077 
•22102 
•22127 
•22152 
-22177 

•15272 
.15288 
.15303 
.15319 
.15334 

.18025 
•18047 
•18068 
•18090 
.18111 

.16212 
.16228 
.16244 
.16260 
.16276 

•19349 
•19372 
•19394 
-19417 
-  19440 

•17178 
•17194 
.17210 
•17227 
.17243 

.20740 
•20764 
•20788 
•20812 
-20836 

•18168 
•18185 

-18202 
.18218 
-18235 

.  22202 
•22227 
•22252 
.22277 
-22302 

.15350 
-15365 
.15381 
-15396 
-15412 

-18133 
.18155 
.18176 
•18198 
.18220 

•16292 
•16308 
-16324 
.16340 
-16355 

.19463 
.19485 
.19508 
-19531 
-19554 

.17259 
.17276 
-17292 
.17308 
•17325 

.20859 
•20883 
-20907 
.20931 

.20955 

-18252 
-18269 
-18286 
-18302 
-18319 

•22327 
.22352 
•22377 
.  22402 
-  22428 

.15427 
•15443 
.15458 
.15474 
.15489 

.18241 
•18263 
•18285 
.18307 
.18328 

-16371 
-16387 
-16403 
-16419 
-16435 

.19576 
•19599 
.19622 
-19645 
-19668 

.17341 
•17357 
•17374 
-17390 

-17407 

.20979 
-21003 
•21027 
-21051 
-21075 

-18336 
-18353 
-18369 
-18386 

-18403 

.22453 
.22478 
.22503 
•22528 

•22554 

.15505 
-15520 
-15536 
-15552 
-15567 

-18350 
-18372 
.18394 
•18416 
.18437 

.16451 
-16467 
-16483 
.16499 
.16515 

-19691 
.19713 
.19736 
.19759 
.19782 

-17423 
-17439 
-17456 
•17472 
.17489 

-21099 
•21123 
-21147 
-21171 
-21195 

•18420 
•18437 
-18454 
-18470 
-18487 

-22579 
•22604 
•22629 
-22655 
-22680 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 

-•15583 
-15598 
-15614 
-15630 
.15645 

.18459 
.18481 
•18503 
.18525 
.18547 

-  16531 
•16547 
•16563 
•16579 
-16595 

.19805 
.19828 
•19851 
.19874 
-19897 

•17505 
-17522 
-17538 
.17554 
.17571 

•21220 
•21244 
•21268 
•21292 
-21316 

•18504 
-18521 
-18538 
-18555 
.18572 

.  22706 
-22731 
.22756 
-22782 
•22807 

30 

31 
32 
33 
34 

.15661 
•15676 
•15692 
•15708 
•15723 

.18569 
.18591 
.18613 
.18635 
-18657 

.16611 
-16627 
•16644 
•16660 
-16676 

.19920 
•19944 
•19967 
•19990 
-20013 

•17587 
•17604 
-17620 
•17637 
-17653 

.21341 
•21365 
-21389 
-21414 
-21438 

-18588 
•18605 
-18622 
-18639 
-18656 

•22833 
-22858 
•22884 
.  22909 
.22935 

30 

31 
32 
33 
34 

35 
36 
87 
38 
39 

.15739 
-15755 
-15770 
•15786 

.15802 

-18679 
.18701 
.18723 
.18745 

•18767 

-16692 
-16708 
-16724 
-16740 
-16756 

•20036 
•20059 
•20083 
•20106 
•20129 

-17670 
-17686 
-17703 
-17719 
-17736 

•21462 
.21487 
.21511 
-21535 

-21560 

•18673 
-18690 
-18707 
-18724 
.18741 

-22960 
-22986 
•23012 
-23037 

.23063 

35 
36 
37 
38 
_39 
4O 
41 
42 
43 
44 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

.15818 
-15833 
•15849 
-15865 
.15880 

.18790 
.18812 
•18834 
-18856 
.18878 

•16772 
-16788 
-16805 
-16821 
-16837 

-20152 
-20176 
-20199 
.20222 
-20246 

•17752 
.17769 
•17786 
.17802 
.17819 

-21584 
-21609 
-21633 
-21658 
-21682 

-18758 
-18775 
•18792 
•18809 
-18826 

.23089 
•23114 
.23140 
-23166 
•23192 

•15896 
.15912 
•15928 
-15943 
•15959 

-18901 
-18923 
-18945 
-18967 
-18990 

-16853 
-16869 
-16885 
-16902 
-16918 

-20269 
-20292 
-20316 
-20339 
-20363 

-17835 
-17852 
-17868 
-17885 
-17902 

•21707 
.21731 
•21756 
•21781 
-21805 

-18843 
•18860 
-18877 
•18894 
•18911 

.23217 
.23243 
-23269 
.23295 
-23321 

45 
46 
47 
48 
49 
5O 
51 
52 
53 
_5J 
55 
56 
57 
58 
59 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

•15975 
•15991 
.16006 
-16022 
•18038 

-19012 
.19034 
.19057 
•19079 
-19102 

-16934 
-16950 
-16966 
-16983 
-16999 

.20386 
•20410 
•20433 
.20457 
-  20480 

-17918 
.17935 
-17952 
•17968 
-17985 

•21830 
•21855 
•21879 
•21904 
•21929 

-18928 
.18945 
•18962 
•18979 
18996 

.23347 
.23373 
.23399 
.  23424 
-23450 

•16054 
-16070 
•16085 
.16101 
.16117 

.19124 
.19146 
.19169 
.19191 
-19214 

.17015 
.17031 
•17047 
•17064 
•17080 

-20504 
-20527 
•20551 
-20575 
-20598 

-18001 
•18018 
•18035 
.18051 
-18068 

.21953 
.21978 
.22003 
.22028 
-22053 

19013 
19030 
19047 
19064 
19081 

.23476 
.23502 
-23529 
-23555 
•23581 

60 

•16133 

.19236 

-17096 

.20622 

-18085 

.22077 

19098 

•23607 

60 

237 


TABLE  11— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS 
36°  37°  38°  39° 


/ 

o 

2 
8 

~6~ 

6 

8 

G 
1O 

11 
12 
13 
li- 
15 
16 
17 
18 
-L9_ 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39_ 
40 
41 
42 
43 
44 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

» 

~0 

2 

_! 

5 

8 
9 
10 

11 
12 
13 
-I* 
15 
16 
17 
18 
19 

•19098 
•19115 
•19133 
.19150 
-19167 

.23607 
•23633 
.23659 
.23685 
•23711 

.20136 
-20154 
.20171 
•20189 
•20207 

.25214 
•25241 
.25269 
.25296 
•25324 

.21199 
.21217 
.21235 
•21253 
.21271 

.26902 
.26931 
•26960 
.26988 
.27017 

.22285 
•22304 
•22322 
.22340 
•22359 

.28676 
-28706 
-28737 
-28767 
•28797 

.19184 
.19201 
.19218 
.19235 
.19252 

.23738 
.23764 
.23790 
.23816 
.23843 

.20224 
.20242 
.20259 
.20277 
.20294 

.25351 
•25379 
•25406 
.25434 
.25462 

•21289 
•21307 
•21324 
.21342 
.21360 

•27046 
.27075 
•27104 
•27133 
•27162 

•22377 
•22395 
•22414 
•22432 
.  22450 

•28828 
.28858 
.28889 
.28919 
•28950 

.19270 
.19287 
•19304 
.19321 
•19338 

.23869 
.23895 
.23922 
.23948 
.23975 

•20312 
.20329 
.20347 
•20365 
.20382 

.25489 
.25517 
.25545 
.25572 
.25600 

•21378 
.21396 
.21414 
.21432 
•21450 

.27191 
•27221 
.27250 
.27279 
•27308 

.22469 
-22487 
-22506 
•22524 
.22542 

.28980 
•29011 
.29042 
•29072 
•29103 

.19356 
.19373 
.19390 
•19407 
-19424 

.24001 
.24028 
.24054 
.24081 
.24107 

•20400 
.20417 
.20435 
.20453 
.20470 

.25628 
.25656 
.25683 
.25711 
.25739 

•21468 
•21486 
.21504 
.21522 
.21540 

.27337 
.27366 
.27396 
.27425 
-27454 

•22561 
.22579 
-22598 
•22616 
•22634 

.29133 
.29164 
.29195 
.29226 
•29256 

•19442 
•19459 
•19476 
•19493 
.19511 

.24134 
-24160 
.24187 
.24213 
.24240 

.20488 
.20506 
•20523 
•20541 
.20559 

•25767 
.25795 
.25823 
.25851 
.25879 

.21558 
.21578 
•21595 
•21613 
.21631 

•27483 
.27513 
.27542 
.27572 
.27601 

.22653 
•22671 
•22690 
•22708 
.22727 

•29287 
•29318 
-29349 
.29380 
.29411 

20 

21 
22 
23 
24 
25 
26 
27 
28 
_29 
30 
31 
32 
33 
34 

.19528 
•19545 
•19562 
.19580 
•19597 

.24267 
.24293 
.  24320 
.24347 
•24373 

•20576 
•20594 
.20612 
.20629 
.20647 

.25907 
.25935 
•25963 
.25991 
.26019 

•21649 
.21667 
.21685 
•21703 
.21721 

.27630 
.27660 
•27689 
.27719 
.27748 

•22745 
.22764 
.22782 
.22801 
-22819 

-29442 
-29473 
.29504 
.29535 
•29566 

.19614 
.19632 
.19649 
.19666 
-19684 

.  24400 
.24427 
.24454 
.24481 
.24508 

•20665 
.20682 
.20700 
.20718 
•20736 

.26047 
.26075 
.26104 
.26132 
.26160 

.21739 
•21757 
•21775 
.21794 
•21812 

-27778 
-27807 
-27837 
.27867 
.27896 

.22838 
.22856 
22875 
.22893 
•22912 

.29597 
.29628 
.29659 
.29690 
•29721 

.19701 
.19718 
•19736 
•19753 
.19770 

.24534 
.24561 
.24588 
.24615 
.  24642 

•20753 
•20771 
•20789 
.20807 
.20824 

.26188 
.26216 
.26245 
.26273 
•26301 

•21830 
.21848 
.21866 
•21884 

-21902 

-27926 
.27956 
.27985 
•28015 

.28045 

•22930 
•22949 
•22967 
•22986 
.  23004 

.29752 
-29784 
-29815 
-29846 
-29877 

35 
38 
37 
38 
-39 
40 
41 
42 
43 
_44 

45 
46 
47 
48 
49 

•19788 
.19805 
•19822 
.19840 
.19857 

.24669 
.24696 
.24723 
.24750 
•24777 

•20842 
•20860 
•20878 
•20895 
.20913 

.26330 
.26358 
.26387 
•26415 
.  26443 

.21921 
•21939 
-21957 
•21975 
.21993 

•28075 
.28105 
-28134 
.28164 
•28194 

•23023 
.23041 
•23060 
.23079 
.23097 

-  29909 
-29940 
-29971 
.30003 
•30034 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

•19875 
•19892 
•19909 
•19927 
.19944 

.24804 
.24832 
24859 
.24886 
•24913 

.20931 
•20949 
•20967 
•20985 
.21002 

•26472 
.26500 
.26529 
.26557 
•26586 

•22012 
.22030 
•22048 
•22066 
.  22084 
•22103 
•22121 
•22139 
•22157 
•22176 

-28224 
-28254 
•28284 
.28314 
•28344 

•23116 
•23134 
-23153 
•23172 
-23190 

.30066 
.30097 
.30129 
.30160 
•30192 

•19962 
.19979 
•19997 
•20014 
•20032 

.  24940 
.24967 
.24995 
.25022 
-  25049 

•21020 
.21038 
.21056 
•21074 
.21092 

.26615 
.26643 
.26672 
.26701 
.26729 

.28374 
-  28404 
-28434 
•28464 
•28495 

•23209 
-23228 
-23246 
•23265 
-23283 

.30223 
.30255 
•30287 
.30318 
.30350 

50 

51 
52 
53 
54 
55 
56 
57 
58 
-fifl 
60 

55 
56 
57 
58 
M. 

60 

.20049 
•20066 
•20084 
•20101 
-20119 

.25077 
.25104 
.25131 
.25159 
.25186 

•21109 
•21127 
•21145 
•21163 
-21181 

.26758 
.26787 
.26815 
.26844 
.26873 

•22194 
•22212 
•22231 
•22249 
.22267 

-28525 
•28555 
.28585 
.28615 
.28646 

•23302 
•23321 
-23339 
•23358 
-23377 

.30382 
.30413 
.30445 
.30477 
-30509 

•20136 

•25214 

.21199 

.26902 

•22285 

•28676 

•23396 

.30541 

238 


TABLE  11.—  NATURAL  VERSED  SINES  AND  EXTERNAL  SECANTS. 
40°          41°          43°           43° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

O 

1 
2 
3 
4 
5 
6 
7 

9 

•23396 
.23414 
•23433 
.23452 
.23470 

.30541 
.30573 
.30605 
.30636 
.30668 

.24529 
.24548 
•24567 
•24586 
.24605 

,32501 
.32535 
.32568 
.32602 
.32636 

.25686 
•25705 
.25724 
.25744 
.25763 

.34563 
.34599 
.34634 
.34669 
.34704 

.26865 
•26884 
•26904 
.26924 
.26944 

•36733 
.36770 
.36807 
.36844 
.36881 

0 

2 
3 

_1 
5 
6 
7 
8 
9 
1O 
11 
12 
13 
14 
"15 
16 
17 
18 
19 

•23489 

.23508 
.23527 
.23545 
•23564 

.30700 
•30732 
.30764 
.30796 
.30829 

•24625 
.24644 
•24663 
•24682 
.24701 

.32669 
.32703 
.32737 
.32770 
.32804 

.25783 
•25802 
•25822 
.25841 
•25861 

.34740 
.34775 
.34811 
•34846 
•34882 

•26964 
•26984 

•27004 
•27024 
•  27043 

•36919 
.36956 
.36993 
.37030 
•37068 

10 

11 
12 

13 

14 

15 

16 
17 
18 

19 

.23583 
.23602 
.23620 
.23639 
•23658 

.30801 
.30393 
.30925 
.30957 
.30989 

.  24720 
•24739 
.24759 
.24778 
.24797 

.32838 
.32872 
.32905 
.32939 
.32973 

.25880 
•25900 
.25920 
.25939 
•25959 

.34917 
•34953 
.34988 
.35024 
.35060 

27063 
•27083 
•27103 
.27123 
•27143 

•37105 
.37143 
•37180 
.37218 
.37255 

.23677 
•23696 
•23714 
•23733 

-23752 

•31C22 
.31054 
.31086 
•31119 
.31151 

•24816 
•24835 
.24854 
.24874 
.24893 

.33007 
.33041 
.33075 
.33109 

•33143 

.25978 
.25998 
.26017 
.26037 
.26056 

.35095 
.35131 
.35167 
.35203 
.35238 

.27163 
.27183 
.27203 
.27223 
.  27243 

.37293 
.37330 
.37368 
.37406 

•37443 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

•23771 
•23790 
.23808 
.23827 
•23846 

•31183 
.31216 
.31248 
.31281 
.31313 

24912 
24931 
24950 
24970 
24989 

.33177 
.33211 
.33245 
.33279 
.33314 

.26076 
.26096 
.26115 
.26135 
.26154 

.35274 
.35310 
.35346 
.35382 
.35418 

.27263 
.27283 
.27303 
•27323 
.27343 

.37481 
.37519 
•37556 
.37594 
.37632 

2O 

21 
22 
23 
_24 
25 
28 
27 
28 
29 
30 
31 
32 
33 
34 

•23865 
23884 
•23903 
•23922 
23941 

.31346 
.31378 
.31411 
.31443 
.31476 

25008 
25027 
25047 
25066 
25085 

.33348 

.33382 
.33416 
.33451 
.33485 

.26174 
.26194 
.26213 
•26233 
.26253 

.35454 
.35490 
.35526 
.35562 
.35598 

.27363 
27383 
27403 
27423 
.27443 

•37670 
.37708 
.37746 
.37784 
.37822 

.23959 
.23978 
23997 
24016 
24035 

.31509 
•31541 
.31574 
.31607 
.31640 

25104 
25124 
25143 
25162 
25182 

.33519 
.33554 
.33588 
.33622 
.33657 

26272 
26292 
26312 
26331 
26351 

.35634 
.35670 
•35707 
.35743 
•35779 

27463 
27483 
27503 
27523 
27543 

.37860 
.37898 
.37936 
.37974 
•38012 

35 
36 
37 
38 
39 

24054 
24073 
24092 
24111 

24130 

.31672 
.31705 
.31738 
.31771 

.31804 

25201 
25220 
25240 
25259 
25278 

.33691 
.33726 
.33760 
.33795 
.33830 

26371 
26390 
26410 
26430 
26449 

.35815 
•35852 
.35888 

.35924 
.35961 

27563 
27583 
27603 
27623 
27643 

.38051 
.38089 
.38127 
.38165 
.38204 

35 
36 
37 
38 
-39 
40 
41 
42 
43 
44 

40 

41 
42 
43 

44 

24149 
24168 
24187 
24206 
24225 

.31837 
.31870 
.31903 
.31936 
•31969 

25297 
25317 
25336 
25356 
25375 

.33864 
.33899 
.33934 
.33968 
.34003 

26469 
26489 
26509 

26528 
26548 

.35997 
.36034 
.36070 
.36107 
.36143 

27663 
27683 
27703 
27723 
27743 

•38242 
.38280 
.38319 
.38357 
.38396 

,  45 
46 

:•  47 

48 

49 

24244 
24262 
24281 
24300 
24320 

.32002 
.32035 
.32068 
.32101 
.32134 

25394 
25414 
25433 
25452 
25472 

.34038 
.34073 
.34108 
.34142 
.34177 

26568 
26588 
26607 
26627 
26647 

.36180 

.36217 
.36253 
.36290 
•36327 

27764 
27784 
27804 
27824 
27844 

.38434 
.38473 
.38512 
.38550 
.38589 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

5O 

51 
52 
53 
54 

24339 
24358 
24377 
24396 
24415 

.32168 
.32201 
•32234 
.32267 
.32301 

25491 
25511 
25530 
25549 
25569 

.34212 
.34247 
•34282 
•34317 
•34352 

26667 
26686 
26706 
26726 
26746 

.36363 
.36400 
.36437 
.36474 
.36511 

27864 
27884 
27905 
27925 
27945 

.38628 
.38666 
.38705 
.38744 
.38783 

55 
56 
57 
58 
59 

24434 
24453 
24472 
24491 
24510 

.32334 
.32368 
.32401 
•32434 
.32468 

25588 
25608 
25627 
25647 
25666 

•34387 

.34423 
.34458 
.34493 
.34528 

26766 
26785 
26805 
26825 
26845 

.36548 
•36585 
.36622 
.36669 
.36696 

27965 
27985 
28005 
28026 
28046 

.38822 
.38860 
.38899 
.38938 
.38977 

60 

24529 

.32501 

25686 

.34563 

26865 

.36733 

28066 

.39016 

60 

239 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
44°  45°  46°  47° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

0 

1 

2 
3 
4 

.28066 
.28086 
.28106 
.28127 
.28147 

.39016 
.39055 
.39095 
.39134 
.39173 

•29289 
-29310 
•29330 
•29351 
•29372 

.41421 
.41463 
.41504 
.41545 
.41586 

.30534 
.30555 
.30576 
•30597 
.30618 

•43956 
.43999 
.44042 
.44086 
-44129 

•31SOO 
•31821 
•31843 
•31864 
•311*5 

•46628 
•46674 
.46719 
•46765 
-46811 

0 

1 
2 
3 
4 

5 
6 
7 
8 
9 

.28167 
.28187 
•28208 
.28228 
.28248 

.39212 
.39251 
.39291 
.39330 
•39369 

•29392 
.29413 
.29433 
29454 
.29475 

.41627 
.41669 
.41710 
•41752 
•41793 

•30639 
.30660 
•30681 
.30702 
•30723 

.44173 
.44217 
•44260 
•44304 
.  44347 

•31907 
•31928 
•31949 
•31971 
•31992 

•46857 
.46903 
•46949 
•46995 
47041 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.28268 
.28289 
•28309 
•28329 
.28350 

.39409 
.39448 
.39487 
.39527 
.39566 

.29495 
.29516 
.29537 
•29557 
•29578 

.41835 
.41876 
.41918 
.41959 
•42001 

.30744 
.30765 
.30786 
.30807 
.30828 

•44391 
.44435 
.44479 
.44523 
•44567 

•32013 
•32035 
•32C56 
-32077 
•32099 

.47087 
.47134 
•47180 
.47226 
47272 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

.28370 
.28390 
.28410 
.28431 
.28451 

.39606 
•39646 
.39685 
.39725 
•39764 

.29599 
.29619 
.29640 
.29661 
•29681 

.42042 
.42084 
.42126 
.42168 
.42210 

.30849 
.30870 
.30891 
.30912 
-30933 

•44610 
.44654 
.44698 
.44742 
-44787 

•32120 
-32141 
-32163 
-32184 
•32205 

.47319 
.47365 
.47411 
•47458 

-47504 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

.28471 
.28492 
.28512 
.28532 
.28553 

•39804 
.39844 
.39884 
.39924 
.39963 

29702 
.29723 
.29743 
•29764 
-29785 

.42251 
.42293 
.42335 
.42377 
.42419 

-30954 
.30975 
-30996 
-31017 
-31038 

.44831 
.44875 
.44919 
.44963 
.45007 

-32227 
•32248 
•32270 
•32291 
•32312 

.47551 
•47598 
.47644 
.47691 
-47738 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

•28573 
•28593 
.28614 
.28634 
.28655 

•40003 
.40043 
.40083 
.40123 
.40163 

,29805 
.29826 
.29847 
•29868 
.29888 

.42461 
.42503 
.42545 
.42587 
.42630 

-31059 
.31080 
.31101 
•31122 
-31143 

.45052 
.45096 
.45141 
.45185 
.45229 

32334 
-32355 
•32377 
•32398 
-32420 

•47784 
.47831 
•47878 
•47925 
-47972 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

.28675 
.28695 
•28716 
•28736 
•28757 

.40203 
.40243 
.40283 
.40324 
.40364 

•29909 
.29930 
•29951 
•29971 
.29992 

.42672 
.42714 
.42756 
.42799 
.42841 

•11165 
•31186 
-31207 
-31228 
.31249 

.45274 
.45319 
.45363 
.45408 
-45452 

•32441 
-32462 
•32484 
•32505 
•32527 

•48019 
•48066 
.48113 
.48160 
•48207 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

.28777 
•28797 
•28818 
.28838 
.28859 

.40404 
.40444 
•40485 
.40525 
•405P5 

.30013 
•30034 
•30054 
•30075 
•30096 

•42883 
.42926 
.42968 
.43011 
.43053 

•31270 
-31291 
-31312 
-31334 

-31355 

.45497 
.45542 
.45587 
•45631 
-45676 

-32548 
-32570 
-32591 
-32613 

•32634 

•48254 
•48301 
-48349 
•48396 
•48443 

35 
36 
37 
38 

39 

40 

41 
42 
43 
44 

.28879 
•28900 
.28920 
.28941 
.28961 

.40606 
.  40646 
.40687 
.40727 
.40768 

•30117 
•30138 
•30158 
.30179 
.30200 

.43096 
.43139 
.43181 
.43224 
.43267 

-31376 
-31397 
•31418 
•31439 
.31461 

•45721 
•45766 
.45811 
.45856 
.45901 

•32656 
-32677 
-32699 
-32720 
-32742 

•48491 
-48538 
•48586 
•48633 
•48681 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 

•28981 
•29002 
•29022 
•29043 
.29063 

.40808 
.40849 
.40890 
.40930 
.40971 

•30221 
•30242 
•30263 
•30283 
.30304 

.43310 
.43352 
.43395 
.43438 
-43481 

•31482 
•31503 
•31524 
•31545 
-31567 

.45946 
.45992 
.46037 
-46082 
46127 

•32763 
-32785 
-32806 
-32828 
-  32849 

•48728 
.48776 
•48824 
•48871 
-48919 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•29084 
.29104 
..'39125 
.29145 
•29166 

.41012 
.41053 
.41093 
.41134 
.41175 

•30325 
•30346 
-30367 
•30388 
.  30409 

.43524 
.43567 
.43610 
.43653 
.43696 

•31588 
•31609 
•31630 
•31651 
•31673 

•46173 
•46218 
•46263 
•46309 
•46354 

-32871 
•32893 
•32914 
32936 
3?957 

•48967 
•49015 
•49063 
.49111 
.49159 

50 

51 

52 
53 
54 

55 
56 
57 
58 
59 

.29187 
•29207 
•29228 
•29248 
29269 

.41216 
.41257 
.41298 
.41339 
.41380 

•30430 
-30451 
•30471 
.30492 
-30513 

.43739 
.43783 
.43826 
.43869 
.43912 

•31694 
31715 
31736 
31758 
31779 

.46400 
.46445 
•46491 
.46537 
-46582 

32979 
33001 
33022 
33044 
330P5 

.49207 
•49255 
.49303 
.49351 
.4P399 

55 
56 
57 
58 
59 

60 

•29289 

.41421 

-30534 

.43956 

31800 

.46628 

33087 

•49448 

60 

240 


TABLE  11— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANT3. 
48*  49°  50°  51° 


• 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

o 

1 

2 
3 

4 

33087 
33109 
33130 
33152 
33173 

.49448 
.49496 
.49544 
.49593 
.49641 

34394 
34416 
34438 
34400 
34482 

•52425 
.52476 
.52527 
.52579 
-52630 

35721 
35744 
35766 
35788 
35810 

•55572 
•55626 
.55680 
.55734 
.55789 

37068 
37091 
37113 
37136 
37158 

•58902 
•58959 
•59016 
.59073 
-59130 

0 

1 
2 
3 

4 

5 
6 
7 
8 
9 
10 
11 
12 
13 
14 

33195 
33217 
33238 
33260 
33282 

•49690 
.49738 
.49787 
.49835 
.49884 

34504 
34526 
34548 
34570 
34592 

.52681 
.52732 
.52784 
.52835 
.52886 

35833 
35855 
35877 
35900 
35922 

.55843 
.55897 
.55951 
.56005 
.  56.060 

37181 
37204 
37226 
37249 
37272 

-59188 

-59245 
•59302 
.59360 
-59418 

5 
6 
7 
8 
9 

33303 
33325 
33347 
33368 
33390 

.49933 
.49981 
.50030 
.50079 
.50128 

34614 
34636 
34658 
34680 
34702 

.52938 
.52989 
.53041 
.53092 
.53144 

35944 
35967 
35989 
36011 
36034 

.56114 
.56169 
.56223 
.56278 
.56332 

37294 
37317 
37340 
37362 
37385 

.59475 
.59533 
•59590 
.59648 
.59706 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

33412 
33434 
33455 
33477 
33499 

.50177 

.50226 
.50275 
.50324 
.50373 

34724 
34746 
34768 
34790 
34815? 

.53196 
.53247 
.53299 
.53351 
.53403 

36056 
36078 
36101 
36123 
36146 

.56387 

.56442 
.56497 
.56551 
.56606 

37408 
37430 
37453 
37476 
37498 

.59764 
.59822 
.59880 
.59938 
.59996 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

.  25 
26 
27 
28 

29 

33520 
33542 
33564 
33586 
33607 

•50422 
.50471 
.50521 
.50570 
.50619 

34834 
34856 
34878 
34900 
34923 

.53455 
.53507 
.53559 
.53611 
.53663 

36168 
36190 
36213 
36235 
36258 

•56661 
.56716 
.56771 
.56826 
.56881 

37521 
37544 
37567 
37589 
37612 

.60054 
.60112 
.60171 
.60229 
.60287 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 

33629 
33651 
33673 
33694 
33716 

•50669 
.50718 
•50767 
.50817 
.50866 

34945 
34967 
34989 
35011 
35033 

.53715 
.53768 
.53820 
.53872 
.53924 

36280 
36302 
36325 
36347 
36370 

.56937 
.56992 
.57047 
.57103 
.57158 

37635 
37658 
.37680 
.37703 
.37726 

.60346 
.60404 
.60463 
.60521 
-60580 

30 

31 
32 
33 
!  34 

33738 
33760 
33782 
33803 
33825 

•50916 
•50966 
•51015 
.51065 
.51115 

35055 
35077 
35099 
35122 
35144 

.53977 
.  54029 
•54082 
.54134 
.54187 

36392 

36415 
36437 
36460 
36482 

.57213 
-57269 
.57324 
.57380 
.57436 

.37749 
•37771 
-37794 
.37817 
.37840 

.60639 
.60698 
.60756 
.60815 
.60874 

3O 

31 
32 
33 
34 
35 
36 
37 
38 
-39 
4O 
41 
42 
43 
44 
45 
46 
47 
48 
49 

5O 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

35 
36 
37 
38 
39_ 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

33847 
33869 
33891 
33912 
33934 

.51165 
.51215 
.51265 
.51314 
.51364 

35166 
35188 
35210 
35232 

35254 

.54240 
.54292 
.  54345 
.54398 

.54451 

36504 
•36527 
.36549 
•36572 
.36594 

-57491 
.57547 
.57603 
.57659 
.57715 

-37862 
-37885 
-37908 
-37931 

.37954 

.60933 
.60992 
.61061 
.61111 
-61170 

33958 
•33978 
•34000 
•34022 
•34044 

.51415 
.51465 
.51515 
.51565 
.51615 

35277 
35299 
35321 
35343 
35365 

.54504 
.54557 
.54610 
.54663 
.54716 

.36617 
•36639 
•36662 
•36684 
.36707 

•57771 
.57827 
.57C83 
.57939 
.57995 

.37976 
.37999 
.38022 
-38045 
-38068 

.61229 
.  61288 
.61348 
•61407 
.61467 

•34065 
-34087 
•34109 
•34131 
-34153 

.51665 
.51716 
.51766 
.51817 
.51867 

35388 
35410 
35432 
.35454 
•35476 

.54769 
.54822 
.54876 
.54929 
.54982 

•36729 
•36752 
-36775 
•36797 
-36820 

.58051 
.58108 
.58164 
.58221 
.58277 

-38091 
-38113 
•38136 
•38159 
.38182 

.61526 
.61586 
.61646 
.61705 
.61765 

50 

51 
52 
53 
54 

-34175 
•34197 
-34219 
•34241 
•  34262 

.51918 
.51968 
.52019 
.52069 
.52120 

-35499 
•35521 
•35543 
.35565 
.35588 

.55036 
.55089 
.55143 
.55196 
.55250 

.36842 
•36865 
•36887 
.36910 
.36932 

.58333 
.58390 
.  58447 
.58503 
.58560 

.38205 
•38228 
•38251 
-38274 
.38296 

-  61825 
.61885 
.61945 
.62005 
.62065 

'55 
56 
57 
58 
59 
60 

•34284 
•34306 
•34328 
•34350 
-34372 

.52171 
.52222 
.52273 
.52323 
.52374 

.35610 
•35632 
•35654 
•35677 
•35699 

.55303 
.55357 
.55411 
.55465 
.55518 

.36955 
-36978 
.37000 
-37023 
-37045 

.58617 
.58674 
.58731 
.58788 
.58845 

-38319 
.38342 
.38365 
.38388 
.38411 

.62125 
.62185 
.62246 
.62306 
-62366 

•34394 

.52425 

.35721 

.55572 

-37068 

.58902 

•38434 

.62427 

241 

TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANT& 
53°  53°  54°  55° 


' 

Vers. 

Ex.  sec 

Vers. 

Ex.  sec 

Vers. 

Ex.  sec 

Vers. 

Ex.  sec 

I 

.  38434 
.38457 
.38480 
.38503 
•38526 

.62427 
.  62487 
.62548 
.62609 
•62669 

.39819 

•39842 
•39865 
•39888 
•39911 

.66164 
.66228 
.66292 
.66357 
•66421 

•41221 
•41245 
•41269 
•41292 
•41316 

.70130 
.70198 
.70267 
•70335 
.  70403 

•42642 
.42666 
•42690 
.42714 
•42738 

.  74345 
•74417 
.  74490 
•74562 
•74635 

0 

2 
3 
4 

1 

e 

1 

8 

9 

•38549 
•38571 
•38594 
•38617 
.38640 

.62730 
.62791 
.62852 
•62913 
.62974 

•39935 
•39958 
.39981 
•40005 
.40028 

•66486 
•66550 
.66615 
.66679 
•66744 

•41339 
•41363 
•41386 
•41410 
•41433 

.  70472 
.  70540 
.  70609 
•70677 
•70746 

•42762 
•42785 
•42809 
•42833 
•42857 

•74708 
.  74781 
.74854 
.74927 
•75000 

5 
6 
7 
8 
g 

10 

11 
12 
13 
14 

•38663 
.38686 
.38709 
•38732 
•38755 

.63035 
.63096 
.63157 
.63218 
.63279 

•40051 
•40074 
.40098 
•40121 
.40144 

.66809 
.66873 
•66938 
.67003 
•67068 

.41457 
•41481 
•41504 
•41528 
-41551 

•  70815 
.70884 
.70953 
.  71022 
•71091 

•42881 
.42905 
•42929 
•42953 
•42976 

.75073 
.75146 
.75219 
.75293 
•75366 

10 

11 
12 
13 
14 

15 
16 
17 
18 
-L2L 

.38778 
.38801 
.38824 
•38847 
•38870 

.63341 
.63402 
.63464 
.63525 
.63587 

•40168 
•40191 
•40214 
•40237 
•40261 

.67133 
.67199 
.67264 
.67329 

.67394 

•41575 
•41599 
.41622 
•41646 
.41670 

.71160 
•71229 
.71298 
.71368 

•71437 

.43000 
.43024 
•43048 
•43072 

.  75440 
.75513 
•75587 
.75661 
•75734 

15 
16 
17 
18 
19 

20 
21 
22 
23 
24_ 

•38893 
.38916 
.38939 
.38962 
.38985 

.63648 
.63710 
.63772 
.63834 
.63895 

.40284 
•40307 
.40331 
•40354 
.40378 

.67460 
.67525 
.67591 
.67656 
.67722 

•41693 
•41717 
•41740 
-41764 
.41788 

•71506 
.71576 
.71646 
.71715 
•71785 

•43120 
•43144 
•43168 
•43192 
43216 

.75808 
.75882 
.75956 
.76031 
•76105 

20 

21 

22 
23 
24 

25 
26 
27 
28 

39009 
39032 
39055 
39078 
39101 

.63957 
.64019 
.64081 
•64144 
•64206 

.40401 
.40424 
.40448 
•40471 
•40494 

.67788 
.67853 
•67919 
•67985 
•68051 

•41811 
•41835 
•41859 
•41882 
.41906 

•71855 
.71925 
.71995 
.72065 
•72135 

•43240 
•43264 
•43287 
•43311 
•43335 

•76179 
.76253 
.76328 
.76402 
•76477 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

39124 
39147 
39170 
39193 
39216 

.64268 
.64330 
.64393 
.64455 
.64518 

-40518 
•40541 
40565 
40588 
40611 

.68117 
.68183 
•68250 
.68316 
.68382 

•41930 
.41953 
.41977 

•42001 
.42024 

.72205 
.  72275 
.72346 
•72416 
.72487 

•43359 
•43383 
•43407 
•43431 
•43455 

•76552 
.76626 
.76701 
.76776 
•76851 

~3O 

31 
32 
33 
34 

(CO  CO  CO  CO  CO 
kOCO-JCDOl 

39239 
39262 
39286 
39309 
39332 

.64580 
.64643 
.64705 
.64768 
.64831 

40635 
40658 
40682 
40705 
40728 

.  68449 
•68515 
.68582 
•68648 
•68715 

•42048 
•42072 
42096 
42119 
42143 

.72557 
.72628 
.72698 
.72769 
.  72840 

•43479 
•43503 
•43527 
•43551 
.43575 

•76926 
.77001 
.77077 
.77152 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

39355 
39378 
39401 
39424 
39447 

•64894 
•64957 
.65020 
.65083 
•65146 

40752 
40775 
40799 
40822 
40846 

•68782 
.68848 
.68915 
.68982 
•69049 

42167 
42191 
42214 
42238 
42262 

.72911 
•72982 
.73053 
.73124 
.73195 

•43599 
•43623 
•43647 
•43671 
43695 

.77303 
.77378 
.77454 
•77530 
•77606 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 

39471 
39494 
39517 
39540 
39563 

.65209 
.65272 
.65336 
.65399 
•65462 

40869 
40893 
40916 
40939 
40963 

.69116 
.69183 
.69250 
.69318 
.69385 

42285 
42309 
42333 
42357 
42381 

.73267 
.73338 
.  73409 
.73481 
.73552 

43720 

43744 
43768 
43792 
43816 

•77681 
•77757 
.77833 
.77910 
•77986 

45 
46 
47 
48 

49 

50 

51 
52 
53 
54 

39586 
39610 
39633 
9656 
9679 

65526 
65589 
65653 
65717 
65780 

40986 
41010 
41033 
41057 
41080 

.69452 
.69520 
.69587 
.69655 
.69723 

42404 
42428 
42452 
42476 
42499 

.73624 
.73696 
.73768 
.73840 
.73911 

43840 
43864 
43888 
43912 
43936 

.78062 
.78138 
.78215 
.78291 
•78368 

50 

51 

52 
53 
54 

55 
56 
57 
58 
-fiL. 

9702 
9726 
9749 
9772 
9795 

65844 
65908 
65972 
66036 
66100 

41104 
41127 
41151 
41174 
41198 

.69790 
•69858 
.69926 
.69994 
•70062 

42523 
42547 
42571 
42595 
42619 

.73983 
.74056 
.74128 
.  74200 
•74272 

43960 
43984 
44008 
44032 
44057 

.78445 
.78521 
.78598 
.78675 
78752 

55 
56 
57 
58 
59 

60 

9819 

66164 

41221 

70130 

42642 

•74345 

44081 

78829 

60 

242 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
56°  57°  58°  59° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

9 

0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

•44081 
.44105 
•44129 
•44153 
•44177 

.78829 
.78906 
.78984 
.79061 
•79138 

.45536 
•45560 
.45585 
-45609 
.45634 

.83608 
.83690 
.83773 
.83855 
.83938 

•47008 
•47033 
•47057 
.47082 
.47107 

.88708 
.88796 
.88884 
.88972 
.89060 

•48496 
•48521 
•48546 
•48571 
-48596 

.94160 
.94254 
.  94349 
.94443 
.94537 

0 

2 
3 
4 
5 
6 
7 

9 

•44201 

.44225 
•44250 
-44274 
-44298 

.79216 
.79293 
.79371 
.  79449 
.79527 

-45658 
•45683 
-45707 
•45731 
.45756 

.84020 
.84103 
.84186 
.84269 
.84352 

.47131 
.47156 
.47181 
•47206 
.47230 

.89148 
•89237 
.89325 
.89414 
.89503 

•48621 
-48646 
•48671 
.48696 
-48721 

.94632 
.94726 
.94821 
.94916 
.95011 

•44322 
•44346 
.44370 
•44395 
.44439 

.79604 
.79682 
.79761 
.79839 
.79917 

.45780 
-45805 
•45829 
.45854 
.45878 

.84435 
•84518 
•84601 
.84685 
-84768 

•47255 
.47280 
.47304 
.47329 
.47354 

.89591 
.89680 
.89769 
.89858 
.89948 

•48746 
-48771 
.48796 
.48821 
.48846 

.95106 
•95201 
.95296 
.95392 
.95487 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

•44443 
•44467 
•44491 
.44516 

.4454.0 

.79995 
.80074 
•80152 
.80231 
.80309 

.45903 
•45927 
.45951 
•45976 

.46000 

.84852 
.84935 
.85019 
•85103 
•85187 

.47379 
•47403 
.47428 
•47453 
•47478 

.90037 
.90126 
•90216 
.90305 
.90395 

•48871 
•48896 
•48921 
.48946 
.48971 

.95583 
-95678 
.95774 
.95870 
•95966 

20 

21 
22 
23 
24 
25 
28 
27 
28 
29 

•44564 
.44588 
•44612 
.44637 
•44661 

.80388 
.80467 
.80546 
.80625 
.80704 

.46025 
-46049 
•46074 
•46098 
.46123 

•85271 
.85355 
.85439 
•85523 
.85608 

.47502 
.47527 
.47552 
.47577 
.47601 

.90485 
.90575 
.90665 
.90755 
.90845 

.48996 
•49021 
•49046 
•49071 
•49096 

.96062 
.96158 
.96255 
.96351 
.96448 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
3] 
32 
33 
34 
35 
36 
37 
38 
-39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
5O 
51 
52 
53 
54 
55 
56 
57 
58 
59 

•44685 
.44709 
•44734 
•44758 
•44782 

.80783 
.80862 
.80942 
.81021 
.81101 

•46147 
•46172 
-46196 
-46221 
•46246 

.85692 
.85777 
.85861 
.85946 
.86031 

•47626 
•47651 
•47676 
.47701 
•47725 

.90935 
.91026 
.91116 
.91207 
.91297 

•49121 

•49146 
-49171 
-49196 
.49221 

.96544 
.96641 
.96738 
.96835 
.96932 

30 

31 
32 
33 
84 
35 
36 
37 
38 
39 

•44806 
.44831 
.44855 
•44879 
.44903 

.81180 
•81260 
•81340 
.81419 
.81499 

-46270 
•46295 
•46319 
•46344 
-46368 

.86116 
.86201 
.86286 
.86371 
.86457 

•47750 
.47775 
-47800 
•47825 
.47849 

.91388 
.91479 
.91570 
.91661 
•91752 

.49246 
.49271 
•49296 
.49321 
.49346 

.97029 
.97127 
.97224 
.97322 
.97420 

•44928 
•44952 
44976 
•45001 

.45025 

•81579 
.81659 
.81740 
.81820 
.81900 

-46393 
•46417 
-  46442 
•46466 
-46491 

•86542 
.86627 
•86713 
.86799 
•86885 

•47874 
.47899 
.47924 
•47949 
-47974 

.91844 
.91935 
.92027 
.92118 
.92210 

•49372 
•49397 
•49422 
-49447 
-49472 

.97517 
.97615 
.97713 
.97811 
.97910 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

•45049 
-45073 
•45098 
•45122 
-45146 

•81981 
.82061 
•82142 
•82222 
•82303 

•46516 
-46540 
•46565 
•46589 
.46614 

•86970 
.87056 
.87142 
•87229 
•87315 

•47998 

.48023 
•48048 
•48073 
.48098 

.92302 
.92394 
.92486 
.92578 
.92670 

•49497 
•49522 
.49547 
.49572 
.49597 

.98008 
.98107 
.98205 
.98304 
.98403 

-45171 
45195 
-45219 
-45244 
-45268 

•82384 
•82465 
•82546 
.82627 
.82709 

•46639 
•46663 
•46688 
•46712 
.46737 

.87401 
•87488 
•87574 
.87661 
.87748 

•48123 
.48148 
•48172 
.48197 
-48222 

.92762 
•92855 
.92947 
.93040 
.93133 

•49623 
•49648 
•49673 
•49698 
.49723 

.98502 
.98601 
.98700 
.98799 
.98899 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

•45292 
•45317 
•45341 
-45365 
-45390 

•82790 
•82871 
•82953 
.83034 
.83116 

•46762 
•46786 
•46811 
•46836 
•46860 

.87834 
.87921 
.88008 
.88095 
•88183 

.48247 
-48272 
.48297 
•48322 
-48347 

.93226 
.93319 
.93412 
.93505 
.93598 

-49748 
-49773 
-49799 
49824 
49849 

.98998 
.99098 
•99198 
.99298 
.99398 

•45414 
•45439 
-45463 
•45487 
•45512 

.83198 
.83280 
.83362 
.83444 
.83526 

•46885 

•46909 
•46934 
-46959 
•46983 

.88270 
•88357 
•88445 
.88532 
•88620 

-48372 
•48396 
•48421 
-48446 
48471 

.93692 
.93785 
.93879 
.93973 
.94066 

49874 
49899 
49924 
49950 
49975 

.99498 
.99598 
.99698 
•99799 
-99899 

60 

•45536 

.83608 

•47008 

.88708 

.48496 

.94160 

50000 

1.00000 

6O 

243 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
60°  61°  63°  63° 


/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

t 

0 

1 

2 
3 
4 

50000 
50025 
50050 
50076 
50101 

1.00000 
1-00101 
1-00202 
1-00303 
1  -  00404 

51519 
51544 
51570 
51595 
51621 

1-06267 
1-06375 
1.06483 
1-06592 
1-06701 

53053 
53079 
53104 
53130 
-53156 

1.13005 
1.13122 
1-13239 
1-13356 
1-13473 

54601 
54627 
54653 
54679 
54705 

1-20269 
1-20395 
1-20521 
1-20647 
1-20773 

0 

1 
2 
3 
4 

~~5~ 
6 
7 
8 
g 

50126 
50151 
50176 
50202 
50227 

1-00505 
1-00607 
1-00708 
1-00810 
1-00912 

51646 
51672 
51697 
51723 
-51748 

1-06809 
1-06918 
1-07027 
1-07137 
1-07246 

-53181 
•53207 
-53233 
.53258 
.53284 

1-13590 
1.13707 
1-13825 
1-13942 
1.14060 

54731 
.54757 
-  54782 
-54808 
.  54834 

1-20900 
1-21026 
1.21153 
1.21280 
1-21407 

5 
6 
7 
8 
9 

ib~ 

11 

12 
13 
14 

50252 
50277 
50303 
50328 
50353 

1-01014 
1-01116 
1-01218 
1-01320 
1-01422 

-51774 
-51799 
-51825 
-51850 
-51876 

1-07356 
1.07465 
1.07575 
1-07685 
1-07795 

-53310 
•53336 
.53361 
.53387 
-53413 

1-14178 
1.14296 
1-14414 
1-14533 
1-14651 

-  54860 
.54886 
-54912 
•54938 
-54964 

1-21535 
1.21662 
1.21790 
1-21918 
1.22045 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

50378 
50404 
50429 
50454 
50479 

1.01525 
1-01628 
1-01730 
1-01833 
1.01936 

-51901 
•51927 
.51952 
.51978 
.52003 

1-07905 
1.08015 
1.08126 
1.08236 
1-08347 

-53439 
-53464 
-53490 
-53516 
-53542 

1.14770 
1.14889 
1.150C8 
1.15127 
1-15246 

•54990 
-55016 
.  55042 
.55068 
-  55094 

1-22174 
1-22302 
1-22430 
1-22559 
1.22688 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

50505 
-50530 
.50555 
.50581 
.50606 

1-02039 
1-02143 
1-02246 
1.02349 
1.02453 

-52029 
-52054 
-52080 
-52105 
-52131 

1-08458 
1.08569 
1.08680 
1.08791 
1-08903 

-53567 
-53593 
-53619 
-53645 
-53670 

1-15366 
1-15485 
1-15605 
1-15725 
1-15845 

-55120 
-55146 
•55172 
•55198 
.55224 

1-22817 
1-22946 
1-23075 
1-23205 
]  .23334 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

•50631 
.50656 
.50682 
.50707 
.50732 

1-02557 
1-02661 
1-02765 
1-02869 
1-02973 

-52156 
-52182 
.52207 
-52233 
.52259 

1-09014 
1.09126 
1-09238 
1.09350 
1-09462 

.53696 
-53722 
-53748 
-53774 
.53799 

1-15965 
1-16085 
1.16206 
1.16326 
1-16447 

•55250 
•55276 
•55302 
•55328 
.55354 

1-23464 
1-23594 
1-23724 
1-23855 
1-23985 

25 
26 
27 
28 
29 

30 

81 
32 
33 
34 

.50758 
.50783 
-50808 
.50834 
-50859 

1-03077 
1-03182 
1-03286 
1-03391 
1-03496 

.52284 
-52310 
.52335 
-52361 
.52386 

1-09574 
1-09686 
1-09799 
1-09911 
1-10024 

-53825 
-53851 
-53877 
-53903 
.53928 

1-16568 
1.16689 
1.16810 
1.16932 
1-17053 

•55380 
-55406 
•55432 
•55458 
-55484 

1-24116 
1-24247 
1-24378 
1-24509 
1-24640 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

.50884 
-50910 
.50935 
-50960 
.  50986 

1-03601 
1-03706 
1-03811 
1-03916 

1-04022 

.52412 
-52438 
.  52463 
.52489 
.52514 

1-10137 
1-10250 
1-10363 
1-10477 
1-10590 

•53954 
-53980 
-  54006 
-54032 

.  RAO  58 

1-17175 
1.17297 
1-17419 
1.17541 
1.17663 

•55510 
•55536 
•55563 
•55589 
-FF615 

1-24772 
1-24903 
1-25035 
1-25167 

1  .95300 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

•51011 
-51036 
-51062 
.51087 
.51113 

1-04128 
1-04233 
1-04339 
1  -  04445 
1-04551 

-52540 
•52566 
-52591 
•52617 
.52642 

1-10704 
1.10817 
1-10931 
1.11045 
1-11159 

-54083 
-54109 
.54135 
-54161 
-54187 

1-17786 
1-17909 
1.18031 
1-18154 
1-18277 

•55641 
•55667 
-55693 
-55719 
-55745 

1-25432 
1-25565 
1-25697 
1-25830 
1.25963 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

•51138 
•51163 
•51189 
.51214 
-51239 

1-04658 
1-04764 
1.04870 
1-04977 
1-05084 

•52668 
•52694 
•52719 
•52745 
.52771 

1-11274 
1-11388 
1-11503 
1.11617 
1-11732 

-54213 
-54238 
-54264 
-54290 
.54316 

1-18401 
1-18524 
1-18648 
1-18772 
1-18895 

•55771 
•55797 
-55823 
-55849 
-55876 

1-26097 
1-26230 
1-26364 
1-26498 
1-26632 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•51265 
•51290 
•51316 
•51341 
.51366 

1-05191 
1-05298 
1-05405 
1-05512 
1-05619 

.52796 
•52822 
.52848 
.52873 
-52899 

1.11847 
1.11963 
1.12078 
1-12193 
1-12309 

-54342 
•54368 
-54394 
-  54420 
.  54446 

1-19019 
1-19144 
1.19268 
1-19393 
1-19517 

.55902 
-55928 
-55954 
-55980 
-56006 

1-26766 
1-26900 
1-27035 
1-27169 
1-27304 

50 

51 
52 
53 
54 

55 
66 
57 
58 
59 

•51392 
•51417 
.51443 
•51468 
-51494 

1-05727 
1-05835 
1-05942 
1-06050 
1-06158 

•52924 
•52950 
-52976 
.53001 
.53027 

1-12425 
1.12540 
1.12657 
1-12773 
1-12889 

-  54471 
•54497 
-54523 
.  54549 
.54575 

1-19642 
1-19767 
1-19892 
1-20018 
1-20143 

-56032 
.56058 
.56084 
-56111 
56137 

1-27439 
1-27574 
1-27710 
1.27845 
1-27981 

bb 
56 
57 
58 

-§C 

60 

•51519 

1-06267 

.53053 

1-13005 

-54601 

1-20269 

-56163 

1-28117 

60 

244 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
64°  65°  66°  67° 


• 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

r 

o 

1 

2 
3 
4 

•56163 
.56189 
.56215 
.56241 
.56267 

1.28117 
1-28253 
1-28390 
1.28526 
1-28663 

.57738 
.57765 
.57791 
.57817 
.  57844 

1-36620 
1-36768 
1.36916 
1-37064 
1.37212 

-59326 
.59353 
.59379 
.59406 
.59433 

.45859 
.46020 
.46181 
.46342 
-46504 

-60927 
•60954 
.60980 
•61007 
.61034 

•55930 
•56106 
.56282 
.56458 
.56634 

O 
3 

5 
6 
7 
8 
9 

.56294 
.56320 
.56346 
.56372 
.56398 

1.28800 
1-28937 
1-29074 
1-29211 
1.29349 

.57870 
.57896 
•57923 
.57949 
.57976 

1-37361 
1.37509 
1.37658 
1-37808 
1.37957 

.  59459 
.59486 

•59512 
•59539 
.59566 

.46665 
•46827 
1.46989 
1-47152 
1-47314 

-61061 
.61088 
-61114 
•61141 
-61168 

•56811 
-56988 
•57165 
-57342 
.57520 

5 
6 

8 

9 

1O 

11 
12 
13 
14 

.56425 
.56451 
•56477 
•56503 
.56529 

1.29487 
1-29625 
.29763 
.29901 
.  30040 

.58002 
.58028 
.58055 
.58081 
.58108 

1-38107 
1-38256 
1.38406 
1.38556 
1-38707 

.59592 
-59619 
-59645 
-59672 
-59699 

1-47477 
1-47640 
1-47804 
1-47967 
1.48131 

-61195 
-61222 
-61248 
-61275 
-61302 

1-57698 
1.57876 
1-58054 
1-58233 
1.58412 

1O 

11 

12 
13 
14 

15 
16 
17 
18 
19 

•56555 
.56582 
.56608 
.56634 
.56660 

•30179 
.30318 
-30457 
-30596 
-30735 

.58134 
-58160 
-58187 
.58213 
-58240 

1-38857 
1-39008 
1-39159 
1.39311 
1-39462 

-59725 
•59752 
-59779 
-59805 
.59832 

1.48295 
1.48459 
1.48624 
1.48789 
1-48954 

.61329 
-61356 
-61383 
-61409 
.61436 

1.58591 
1.58771 
1-58950 
1-59130 
1-59311 

15 
16 
17 
18 
19. 

20 

21 
22 
23 
24 

.56687 
•56713 
•56739 
.56765 
•56791 

•30875 
•31015 
•31155 
•31295 
-31436 

-58266 
•58293 
.58319 
.58345 
.58372 

1.39614 
1.39766 
1-39918 
1.40070 
1.40222 

-59859 
-59885 
.59912 
-59938 
.59965 

1-49119 
1-49284 
1-49450 
1-49616 
1-49782 

.  61463 
.61490 
.61517 
.61544 
-61570 

1-59491 
1-59672 
1-59853 
1.60035 
1-60217 

20 

21 
22 
23 
24 

25 
2G 
27 
28 
29 

•56818 
56844 
56870 
56896 
56923 

•31576 
•31717 
.31858 
.31999 
.32140 

-58398 
.58425 
.58451 
58478 
58504 

1.40375 
1.40528 
1.40681 
1-40835 
1.40988 

-59992 
-60018 
-60045 
•  60072 
.60098 

1  -  49948 
1-50115 
1-50282 
1  .  50449 
1.50617 

•61597 

•61624 
.61651 
.61678 
.61705 

1-60399 
1-60581 
1-60763 
1-60946 
1-61129 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

56949 
56975 
57001 
57028 
57054 

•32282 

•32424 
.32566 
1-32708 
1-32850 

58531 
58557 
58584 
58610 
58637 

1-41142 
1.41296 
1.41450 
1.41605 
1.41760 

•60125 
•60152 
•60178 
-60205 
.60232 

1.50784 
1.50952 
1.51120 
1.51289 
1-51457 

.61732 
.61759 
-61785 
•61812 
.61839 

1-61313 
1-61496 
1.61680 
1-61864 
1  •  62049 

30 

S 

33 
34 

35 
36 
37 
38 

39_ 

57080 
57106 
57133 
57159 
57185 

1-32993 
1.33135 
1-33278 
1-33422 
1-33565 

58663 
58690 
58716 
58743 

58769 

1-41914 
1.42070 
1.42225 
1.42380 
1-42536 

•60259 
-60285 
•60312 
•60339 
-60365 

1.51626 
1-51795 
1.51965 
1.52134 
1  .  52304 

•61866 
•61893 
-61920 
-61947 
-61974 

1-62234 
1.62419 
1-62604 
1.62790 

1-62976 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

57212 
57238 
•57264 
.57291 
.57317 

1-33708 
1-33852 
1-33996 
1-34140 
1-34284 

-58796 
58822 
58849 
-58875 
.58902 

1.42692 
1-42848 
1.43005 
1.43162 
1-43318 

•60392 
•60419 
-  60445 
•60472 
.  60499 

1.52474 
1.52645 
1.52815 
1.52986 
1.53157 

-62001 
-62027 
-62054 
-62081 
-62108 

1-63162 
1.63348 
1-63535 
1-63722 
1-63909 

40 

41 
42 
43 

44 

45 
46 
47 
48 
49 

•57343 
•57369 
-57396 
.  57422 
.  57448 

1  .  34429 
1.34573 
1-34718 
1-34863 
1-35009 

.58928 
•58955 
-58981 
•59008 
-59034 

1.43476 
1-43633 
1.43790 
1.43948 
1.44106 

.60526 
•60552 
.60579 
.60606 
•60633 

1.53329 
1.53500 
1.53672 
1.53845 
1-54017 

•62135 
-62162 
•62189 
-62216 
-  62243 

1.64097 
1-64285 
1-64473 
•64662 
•64851 

45 
46 
47 
48 

49 

50 

51 
52 
53 
54 

.57475 
-57501 
.57527 
.57554 
-57580 

1.35154 
1.35300 
1.35446 
1-35592 
1-35738 

•59061 
-59087 
•59114 
•59140 
.59167 

1.44264 
1  •  44423 
1.44582 
1-44741 
1-44900 

.60659 
•60886 
.60713 
.  60740 
•60766 

1.54190 
1-54363 
1-54536 
1  -  54709 
1  -  54883 

-  62270 

-62297 
•62324 
.62351 
-62378 

.65040 
•65229 
•65419 
1-65609 
1.65799 

5O 

51 
52 

53 
54 

55 
56 
57 
58 
.59. 

.57606 
•57633 
•57659 
.57685 
.57712 

1-35885 
1-36031 
1-36178 
1-36325 

1-36473 

•59194 
•59220 
-59247 
-59273 
-59300 

1-45059 
1-45219 
1.45378 
1.45539 
1.45699 

•60793 
•60820 
•60847 
-60873 
-60900 

1-55057 
1-55231 
1.55405 
1.55580 
1.55755 

.  62405 
.62431 
-62458 
-62485 

-62512 

1-65989 
1.66180 
1  66371 
1-66563 
1-66755 

55 
56 
57 
58 
59 

60 

•57738 

1-36620 

-59326 

1-45859 

.60927 

1.55930 

.62539 

1-66947 

60 

245 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
68°  69°  70°  71° 


/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

0 

2 
3 

.62539 
.62566 
.62593 
•  62620 
•62647 

1-66947 
1.67139 
1-67332 
1.67525 
1-67718 

•64163 
-64190 
.64218 
-64245 
-  64272 

1-79043 
1.79254 
1.79466 
1-79679 
1-79891 

•65798 
•65825 
•65853 
.65880 
.65907 

1-92380 
1-92614 
1-92849 
1-93083 
1.93318 

-  67443 
•  67471 
-  67498 
-67526 
-67553 

2.07155 
2-07415 
'2-07675 
2-07936 
2-08197 

0 

2 
3 

4 

5 
6 
7 

8 

.  9 

.62674 
.62701 
.62728 
•62755 
.62782 

1-67911 
1.68105 
1-68299 
1.68494 
1-68689 

-64299 
•64326 
•64353 
.64381 
.  64408 

1.80104 
1.80318 
1-80531 
1-80746 
1-80960 

•65935 
-65962 
-65989 
-66017 
•66044 

1-93554 
1-93790 
1.94026 
1-94263 
1-94500 

-67581 
-67608 
-67636 
-67663 
-67691 

2-08459 
2-08721 
2-08983 
2-09246 
2-09510 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

•62809 
•62836 
•62863 
.62890 
•62917 

1.68884 
1-69079 
1.69275 
1.69471 
1-69667 

.  64435 
-  64462 
.  64489 
-64517 
-  64544 

1-81175 
1.81390 
1-81605 
1-81821 
1-82037 

-66071 
-66099 
-66126 
-66154 
-66181 

1-94737 
1-94975 
1-95213 
1-95452 
1.95691 

-67718 
-67746 
-67773 
-67801 
-67829 

2.09774 
2-10038 
2-10303 
2.10568 
2-10834 

10 

11 
12 
13 
34 

15 
16 
17 
18 
19 

•62944 
•62971 
•62998 
•63025 
.63052 

1.69864 
1-70061 
1.70258 
1.70455 
1.70653 

-64571 
.64598 
•64625 
•64653 
.  64680 

1-82254 
1-82471 
1-82688 
1-82906 
1-83124 

-66208 
-66236 
-66263 
-66290 
•66318 

1-95931 
1.96171 
1-96411 
1-96652 
1-96893 

-67856 
-67884 
-67911 
-67939 
-67966 

2-11101 
2-11367 
2-11635 
2.11903 
2-12171 

15 
16 
17 
18 

19 

20 

21 
22 
23 
24 

.63079 
•63106 
•63133 
•63161 
•63188 

1.70851 
1.71050 
1.71249 
1  •  71448 
1.71647 

.  64707 
.  64734 
.64761 
•64789 
.64816 

1-83342 
1-83561 
1.83780 
1-83999 
1-84219 

•66345 
-66373 
•  66400 
•66427 
•66455 

1  97135 
1-97377 
1-97619 
1-97862 
1.98106 

-67994 
-68021 
-68049 
-68077 
-68104 

2-12440 
2-12709 
2-12979 
2-13249 
2-13520 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

•63215 
-63242 
•63269 
•63296 
63323 

1-71847 
1  .  72047 
1.72247 
1  .  72448 
1.72649 

.  64843 
.  64870 
.64898 
.  64925 
•64952 

1.84439 
1.84659 
.84880 
.85102 
•85323 

•66482 
•66510 
•66537 
•66564 
.66592 

1-98349 
1-98594 
1-98838 
1-99083 
1-99329 

•68132 
•68159 
•68187 
•68214 
-  68242 

2-13791 
2-14063 
2-14335 
2-14608 
2-14881 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

63350 
•63377 
63404 
63431 
63458 

1-72850 
1-73052 
1-73254 
1-73456 
1-73659 

-64979 
•65007 
•65034 
•65061 
•65088 

-85545 
-85767 
.85990 
-86213 
1.86437 

•66619 
•66647 
•66674 
•  66702 
•66729 

1-99574 
1-99821 
2-00067 
2-00315 
2-00562 

•  68270 
-68297 
•68325 
•68352 
-68380 

2-15155 
2-15429 
2.15704 
2.15979 
2-16255 

30 

sa 

32 
33 
34 

35 
36 
37 
38 
39 

63485 
63512 
63539 
63566 
63594 

1.73862 
1-74065 
1.74269 
1  -  74473 
1-74677 

65116 
65143 
65170 
65197 

65225 

1.86661 
1-86885 
1-87109 
1-87334 
1-87560 

•66756 
•66784 
•66811 
•66839 
-6R866 

2-00810 
2-01059 
2-01308 
2-01557 
2-01807 

•  68408 
•68435 
-68463 
•68490 
.68518 

2-16531 
2-16808 
2-17085 
2-17363 
2-17641 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

63621 
63648 
63675 
63702 
63729 

1.74881 
1.75086 
1.75292 
1.75497 
1-75703 

65252 
65279 
65306 
65334 
65361 

1-87785 
1-88011 
1-88238 
1-88465 
1-88692 

•66894 
•66921 
•66949 
•66976 
.  67003 

2-02057 
2-02308 
2-02559 
2-02810 
2-03062 

-68546 
•68573 
•68601 
.68628 
-68656 

2-17920 
2-13199 
2-18479 
2-18759 
2-19040 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

63756 
63783 
63810 
63838 
63865 

1.75909 
1-76116 
1.76323 
1.76530 
1.76737 

65388 
65416 
65443 
65470 
65497 

1-88920 
1-89148 
1.89376 
1-89605 
1.89834 

-67031 
•67058 
•67086 
67113 
67141 

2-03315 
2-03568 
2-03821 
2-04075 
2-04329 

.68684 
•68711 
•68739 
•68767 
.68794 

2-19322 
2-19604 
2-19886 
2-20169 
2-20453 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

63892 
63919 
63946 
63973 
64000 

1.76945 
1-77154 
1-77362 
1.77571 
1.77780 

65525 
65552 
65579 
65607 
65634 

.90063 
.90293 
.90524 
•90754 
-90986 

67168 
67196 
67223 
67251 
67278 

2-04584 
2-04839 
2-05094 
2-05350 
2-05607 

-68822 
.  68849 
•68877 
•68905 
68932 

2-20737 
2-21021 
2-21306 
2-21592 
2-21878 

50 

51 

52 
53 
54 

55 
56 
57 
58 
1L. 

64027 
64055 
64082 
64109 
64136 

1.77990 
1-78200 
1-78410 
1.78621 
1-78832 

65661 
65689 
65716 
65743 
65771 

-91217 
-91449 
1-91681 
1.91914 
1-92147 

67306 
67333 
67361 
67388 

67416 

2-05864 
2-06121 
2-06379 
2-06637 
2-06896 

68960 
68988 
69015 
69043 
69071 

2-22165 
2-22452 
2-22740 
2-23028 
2.23317 

55 
56 
57 
58 
59 

60 

64U63 

1.79043 

657.98 

1-92380 

67443 

2-07155 

69098 

2-23607 

60 

246 


TABLE  11.— NATURAL    VERSED    STNES    AND    EXTERNAL    SECANTS. 
72°  73°  74°  75° 


/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

o 

1 

2 
3 

| 

-69098 
.69126 
.69154 
•69181 
.69209 

2-23607 
2.23897 
2-24187 
2-24478 
2-24770 

-70763 
-70791 
-70818 
-70846 
-70874 

2-42030 
2-42356 
2-42683 
2-43010 
2-43337 

-72436 
-72464 
-72492 
.72520 
.  72548 

2-62796 
2.63164 
2-63533 
2-63903 
2-64274 

-74118 
-74146 
-74174 
-74202 
-74231 

2-86370 
2-86790 
2-87211 
2-87633 
2-88056 

O 

1 
2 
3 

5 
6 
7 
8 
9 

.69237 
.69264 
.69292 
.69320 
.69347 

2-25062 
2-25355 
2-25648 
2-25942 
2-26237 

.70902 
.70930 
.70958 
-70985 
.71013 

2-43666 
2-43995 
2-44324 
2-44655 
2.44986 

•72576 
-72604 
•72632 
-72660 
.72688 

2.64645 
2.65018 
2-65391 
2-65765 
2.66140 

-74259 
.74287 
.74315 
.74343 
.74371 

2-88479 
2-88904 
2-89330 
2-89756 
2-90184 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

-69375 
-  69403 
-69430 
.69458 
.69486 

2-26531 
2-26827 
2-27123 
2-27420 
2-27717 

-71041 
-71069 
-71097 
-71125 
-71153 

2-45317 
2-45650 
2-45983 
2-46316 
2-46651 

.72716 
-  72744 
-72772 
-72800 
.72828 

2-66515 
2-66892 
2-67269 
2-67647 
2-68025 

•74399 
.  74427 
•74455 
-  74484 
.74512 

2.90613 
2-91042 
2-91473 
2.91904 
2-92337 

10 

11 

12 
13 
14 

15 
16 
17 
18 
19 

•69514 
.69541 
-69569 
-69597 

-69624 

2-28015 
2-28313 
2-28612 
2-28912 

2-29212 

-71180 
.71208 
-71236 
-71264 

-71292 

2-46986 
2-47321 
2-47658 
2-47995 
2  48333 

.72856 
-72884 
-72912 
-72940 
.72988 

2-68405 
2-68785 
2-69167 
2-69549 
2-69931 

.  74540 
-74568 
-74596 
-74624 
-74652 

2-92770 
2-93204 
2-93640 
2-94076 
2-94514 

15 
16 
17 
18 

19 

20 

21 
22 
23 
24 

-69652 
-69680 
.69708 
.69735 
.69763 

2-29512 
2-29814 
2-30115 
2-30418 
2.30721 

-71320 
-71348 
-71375 
•71403 
.71431 

2-48671 
2-49010 
2-49350 
2-49691 
2-50032 

-72996 
-73024 
-73052 
•73080 
-73108 

2-70315 
2-70700 
2-71085 
2.71471 
2-71858 

-  74680 
-  74709 
-74737 
-74765 
.  74793 

2.94952 
2-95392 
2-95832 
2-96274 
2-96716 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

.69791 
.69818 
-69846 
-69874 
-69902 

2.31024 
2.31328 
2-31633 
2-31939 
2-32244 

-71459 
-71487 
-71515 
.71543 
.71571 

2-50374 
2-50716 
2-51060 
2-51404 
2-51748 

-73136 
-73164 
-73192 
.73220 
-73248 

2.72246 
2-72635 
2-73024 
2-73414 
2-73806 

.  74821 
.  74849 
.74878 
-  74906 
-  74934 

2.97160 
2-97604 
2.98050 
2.98497 
2-98944 

23 
26 
27 
28 

29 

30 

31 
32 
33 
34 

-69929 
-69957 
-69985 
-70013 
-  70040 

2.32551 
2-32858 
2-33166 
2-33474 
2-33783 

•71598 
-71626 
-71654 
-71682 
-71710 

2.52094 
2-52440 
2-52787 
2-53134 
2-53482 

-73276 
-73304 
73332 
-73360 
-73388 

2-74198 
2-74591 
2-74984 
2-75379 
2-75775 

-74962 
-74990 
-75018 
.75047 
.75075 

2-99393 
2-99843 
3-00293 
3-00745 
3-01198 

3O 

31 
32 
33 
34 

35 
36 
37 
38 
19- 

•70068 
•70096 
-70124 
•70151 

-70179 

2-34092 
2-34403 
2-34713 
2-35025 
2-35336 

-71738 
-71766 
-71794 
-71822 
.71850 

2-53831 
2-54181 
2-54531 
2-54883 
2-55235 

-73416 
-  73444 
-  73472 
-73500 
-73529 

2-76171 
2-76568 
2-76966 
2-77365 

2-777R5 

-75103 
-75131 
-75159 
-75187 
.75216 

3-01652 
3.02107 
3-02563 
3-03020 
3-03479 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

-70207 
-70235 
-  70263 
-  70290 
-70318 

2-35649 
2-35962 
2-36276 
2.36590 
2-36905 

-71877 
-71905 
-71933 
-71961 
-71989 

2-55587 
2-55940 
2.56294 
2-56649 
2.57005 

-73557 
-73585 
-73613 
-73641 
-73669 

2-78166 
2-78568 
2-78970 
2-79374 
2-79778 

-  75244 
-75272 
-75300 
-75328 
-75356 

3-03938 
3-04398 
3-04860 
3-05322 
3-05786 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 

-  70346 
-70374 
-  70401 
.  70429 
-70457 

2-37221 
2-37537 
2-37854 
2.38171 
2-38489 

-72017 
-  72045 
-72073 
-72101 
.72129 

2-57361 
2-57718 
2-58076 
2-58434 
2-58794 

-73697 
-73725 
-73753 
-73781 
-73809 

2-80183 
2-80589 
2-80996 
2-81404 
2-81813 

-75385 
-75413 
-  75441 
.75469 
-75497 

3-06251 
3-06717 
3-07184 
3-07652 
3-08121 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

-70485 
-70513 
-70540 
•70568 
.70596 

2-38808 
2-39128 
2-39448 
2.39768 
2.40089 

.72157 
•72185 
.72213 
•72241 
•72269 

2.59154 
2-59514 
2-59876 
2-60238 
2.60601 

-73837 
-73865 
-73893 
-73921 
-73950 

2-82223 
2-82633 
2-83045 
2-83457 
2-83871 

-75526 
-75554 
-75582 
-75610 
-75639 

3-08591 
3-09063 
3-09535 
3-10009 
3-10484 

5O 

51 
52 
53 
54 

55 
56 
57 
58 
59. 

.70624 
•70652 
•70679 
•70707 
-70735 

2-40411 
2-40734 
2-41057 
2-41381 
2-41705 

•72296 
-72324 
-72352 
•72380 
-72408 

2-60965 
2-61330 
2-61695 
2-62061 
2-62428 

-73978 
-74006 
-  74034 
-74062 
-74090 

2-84285 
2-84700 
2-85116 
2-85533 

2-85951 

-75667 
-75695 
-75723 
-75751 
-75780 

3-10960 
3.11437 
3-11915 
3.12394 
3-12875 

55 
56 
57 
58 
59 

60 

•70763 

2-42030 

.72436 

2-62796 

-74118 

2-86370 

-75808 

3-13357 

60 

247 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
76°  77°  78°  79° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

0 

2 
3 
4 

-75808 
•75836 
•75864 
•75892 
•75921 

3-13357 
3-13839 
3.14323 
3-14809 
3-15295 

•77505 
•77533 
•77562 
•77590 
•77618 

3-44541 
3-45102 
3-45664 
3-46228 
3-46793 

.79209 
•79237 
•79266 
•79294 
-79323 

3-80973 
3-81633 
3-82294 
3-82956 
3-83621 

-80919 
•80948 
-80976 
-81005 
•81033 

4-24084 
4-24870 
4-25658 
4-26448 
4-27241 

O 

1 
2 
3 
4 

5 
6 
7 
8 
9 

•75949 
•75977 
•76005 
•76034 
•76062 

3-15782 
3.16271 
3-16761 
3-17252 
3-17744 

•77647 
•77675 
•77703 
•77732 
•77760 

3-47360 
3-47928 
3-48498 
3-49069 
3-49642 

•79351 
.79380 
•79408 
-79437 
-79465 

3.84288 
3-84956 
3.85627 
3-86299 
3-86973 

-81062 
•81090 
•81119 
•81148 
•81176 

4.28036 
4-28833 
4-29634 
4-30436 
4-31241 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

•76090 
•76118 
•76147 
•76175 
•76203 

3-18238 
3.18733 
3-19228 
3.19725 
3.20224 

.77788 
.77817 
.77845 
.77874 
•77902 

3-50216 
3-50791 
3-51368 
3-51947 
3-52527 

-  79493 
-79522 
-79550 
-79579 
-  79607 

3-87649 
3-88327 
3-89007 
3-89689 
3-90373 

.81205 
•81233 
•81262 
-81290 
-81319 

4-32049 
4.32859 
4.33671 
4.34486 
4-35304 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

•76231 
•76260 
•76288 
•76316 

•76344 

3-20723 
3-21224 
3-21726 
3-22229 

3-22734 

.77930 
.77959 
-77987 
•78015 

-78044 

3-53109 
3-53692 
3.54277 
3  -  54863 

3-55451 

-79636 
-79664 
-79693 
.79721 

-79750 

3.91058 
3-91746 
3.92436 
3-93128 

3-93821 

•81348 
•81376 
•81405 
-81433 
.81462 

4-36124 
4-36947 
4-37772 
4-38600 
4-39430 

15 
16 
17 
18 

19 

20 

21 
22 
23 

24 

•76373 
.  76401 
•76429 
•76458 
•76486 

3-23239 
3-23746 
3-24255 
3-24764 
3-25275 

.  78072 
.78101 
.78129 
•78157 
•78186 

3  .  56041 
3-56632 
3.57224 
3.57819 
3-58414 

-79778 
-79807 
-79835 
•79864 
•79892 

3-94517 
3-95215 
3-95914 
3-96616 
3-97320 

•81491 
-81519 
.81548 
.81576 
.81605 

4-40263 
4-41099 
4-41937 
4-42778 
4-43622 

20 

2. 
2i. 
23 
24 

25 
26 
27 
28 
29 

•76514 
•76542 
•76571 
•76599 
.76627 

3.25787 
3-26300 
3-26814 
3-27330 
3-27847 

.  78214 

.  78242 
.  78271 
.  78299 
-78328 

3-59012 
3-59611 
3-60211 
3.60813 
3-61417 

•79921 
•79949 
•79978 
•8C006 
•80035 

3-98025 
3-98733 
3-99443 
4-00155 
4-00869 

•81633 
•81662 
•81691 
•81719 
-81748 

4-44468 
4-45317 
4-46169 
4-47023 
4-47881 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

•76655 
•76684 
.76712 
•  76740 
76769 

3-28366 
3-28885 
3  -  29406 
3-29929 
3-30452 

•78356 
•  78384 
•78413 
.  78441 
.  78470 

3  -  62023 
3-62630 
3-63238 
3  •  63849 
3-64461 

-80063 
-80092 
-80120 
-80149 
-80177 

4-01585 
4-02303 
4-03024 
4-03746 
4-04471 

•81776 
•81805 
-81834 
•31862 
•81891 

4-48740 
4-49603 
4-50468 
4-51337 
4-52208 

30 

31 
32 
33 
34 

35 
36 
37 
38 
2* 

•76797 
•76825 
76854 
76882 
76910 

3-30977 
3-31503 
3-32031 
3-32560 
3-33090 

.  78498 
•78526 
•78555 
.78583 
•78612 

3-65074 
3-65690 
3-66307 
3-66925 

3-67545 

-80206 
-80234 
-80263 
-80291 
.80320 

4-05197 
4-05926 
4-06657 
4-07390 
4.08125 

•81919 
•81948 
•81977 
•82005 
-82034 

4-53081 
4-53958 
4-54837 
4-55720 
4.56605 

35 
36 
37 
38 
-29 

40 

41 
42 
43 
44 

76938 
76967 
76995 
77023 
77052 

3-33622 
3-34154 
3-34689 
3-35224 
3-35761 

•78640 
•78669 
•78697 
78725 
78754 

3-68167 
3-68791 
3.69417 
3  -  70044 
3  -  70673 

•80348 
•80377 
-80405 
•  80434 
•80462 

4-08863 
4-09602 
4-10344 
4-11088 
4-11835 

•82063 
•82091 
•82120 
•82148 
-82177 

4-57493 
4-58383 
4-59277 
4-60174 
4-61073 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

77080 
77108 
77137 
77165 
77193 

3-36299 
3-36839 
3-37380 
3-37923 
3-38466 

78782 
78811 
78839 
78868 
78896 

3-71303 
3-71935 
3-72569 
3-73205 
3-73843 

-80491 
•80520 
80548 
80577 
80605 

4-12583 
4-13334 
4-14087 
4-14842 
4-15599 

•82206 
•82234 
-82263 
82292 
82320 

4-61976 
4-62881 
4-63790 
4-64701 
4-65616 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

77222 
77250 
77278 
77307 
77335 

3-39012 
3-39558 
3-40106 
3-40656 
3-41206 

78924 
78953 
78981 
79010 
79038 

3  •  74482 
3-75123 
3-75766 
3-76411 
3-77057 

80634 
80662 
80691 
80719 
80748 

4-16359 
4-17121 
4-17886 
4-18652 
4-19421 

82349 
82377 
82406 
82435 
82463 

4-66533 
4-67454 
4-68377 
4-69304 
4.70234 

50 

51 

52 
53 
54 

55 
56 
57 
58 
19 

77363 
77392 
77420 
77448 
77477 

3-41759 
3-42312 
3-42867 
3-43424 
3.43982 

79067 
79095 
79123 
79152 
79180 

-77705 
-78355 
-79007 
-79661 
-80316 

80776 
80805 
80833 
80862 
80891 

4.20193 
4-20966 
4-21742 
4-22521 
4-23301 

82492 
82521 
82549 
82578 
82607 

4-71166 
4-72102 
4-73041 
4-73983 
4-74929 

55 
56 
57 
58 
59 

60 

77505 

8-44541 

79209 

.80973 

80919 

4-24084 

82635 

4-75877 

60 

248 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
80°  81°  82°  83° 


/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

t 

0 

2 
3 

4 

-82635 
•82664 
.82692 
.82721 
-82750 

4-75877 
4-76829 
4.77784 
4-78742 
4.79703 

.84357 
.84385 

.84414 
.84443 
.  84471 

5-39245 
5-40422 
5.41602 
5.42787 
5-43977 

-86083 
-86112 
-86140 
-86169 
-86198 

6-18530 
6-20020 
6.21517 
6.23019 
6-24529 

.87813 
.87842 
-87871 

-87900 
.87929 

7.20551 
7.22500 
7.24457 
7-26425 
7-28402 

O 

2 
3 

5 
6 
7 
8 
9 

•82778 
-82807 
•82836 
•82864 
-82893 

4.80667 
4-81635 
4-82606 
4-83581 
4.84558 

.84500 
-84529 
-84558 
.84586 
.84615 

5-45171 
5-46369 
5.47572 
5.48779 
5-49991 

-86227 
-86256 
-86284 
•86313 
.86342 

6-26044 
6-27566 
6-29095 
6-30630 
6-32171 

•87957 
.87986 
•88015 
.  88044 
•88073 

7-30388 
7-32384 
7-34390 
7-36405 
7-38431 

5 

6 
7 
8 
9 

10 

11 
12 
13 
14 

•82922 
•82950 
-82979 
.83008 
•83036 

4.85539 
4-86524 
4-87511 
4-88502 
4-89497 

.84644 
.84673 
.84701 
.84730 
.84759 

5.51208 
5-52429 
5-53655 
5.54886 
5-56121 

-86371 
•86400 
-86428 
-86457 
.86486 

6-33719 
6-35274 
6-36835 
6-38403 
6-39978 

.88102 
•88131 
-88160 
.88188 
.88217 

7-40466 
7-42511 
7-44566 
7-46632 
7-48707 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

•83065 
-83094 
-83122 
-83151 
.83180 

4.90495 
4-91496 
4-92501 
4-93509 
4-94521 

.84788 
.84816 
.84845 
.84874 
.  84903 

5.57361 
5-58606 
5.59855 
5-61110 
5-62369 

-86515 
-86544 
-86573 
-86601 
-86630 

6-41560 
6-43148 
6-44743 
6-46346 
6-47955 

-88246 
•88275 
-88304 
-88333 
•88362 

7-50793 
7-52889 
7-54996 
7-57113 

7-59241 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

•83208 
•83237 
-83266 
•83294 
83323 

4-95536 
4-96555 
4-97577 
4-98603 
4-99633 

-84931 
-84960 
-84989 
.85018 
.85046 

5.63633 
5-64902 
5-66176 
5-67454 
5-68738 

-86659 
-86688 
-86717 
•86746 
.86774 

6-49571 
6-51194 
6-52825 
6-54462 
6-56107 

-88391 

-88420 
•88448 
-88477 
.88506 

7-61379 
7-63528 
7-65688 
7-67859 
7-70041 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

83352 
83380 
83409 
83438 
83467 

5-00666 
5-01703 
5.02743 
5-03787 
5-04834 

.85075 
.85104 
.85133 
85162 
.85190 

5-70027 
5-71321 
5-72620 
5-73924 
5-75233 

•86803 
.86832 
•86861 
.86890 
.86919 

6-57759 
6-59418 
6-61085 
6-62759 
6  -  64441 

•88535 
•88564 
-88593 
-88622 
-88651 

7-72234 
7.74438 
7.76653 
7.78880 
7.81118 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

83495 
83524 
83553 
83581 
83610 

5-05886 
5-06941 
5-08000 
5-09062 
5-10129 

.85219 
-85248 
-85277 
-85305 
-85334 

5-76547 
5-77866 
5-79191 
5-80521 
5-81856 

-86947 
-86976 
-87005 
-87034 
-87063 

6.66130 
6-67826 
6-69530 
6-71242 
6-72962 

-88680 
-88709 
•88737 
.88766 
.88795 

7.83367 
7-85628 
7-87901 
7.90186 
7-92482 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39- 

83639 
83667 
83696 
-83725 
•83754 

5-11199 
5.12273 
5-13350 
5-14432 
5-15517 

-85363 
.85392 
.85420 
.85449 
.85478 

5-83196 

5-84542 
5-85893 
5-87250 
5-88612 

.87092 
.87120 
-87149 
-87178 
-87207 

6-74689 
6-76424 
6-78167 
6-79918 

6-81677 

-88824 
-88853 
.88882 
.88911 
-  88940 

7,94791 
7-97111 
7-99444 
8-01788 
8.04146 

35 
36 
37 
38 
-39 

40 

41 
42 
43 
44 

-83782 
-83811 
•83840 
-83868 
-83897 

5-16607 
5-17700 
5-18797 
5.19898 
5-21004 

-85507 
-85536 
-85564 
-85593 
-85622 

5-89979 
5.91352 
5.92731 
5.94115 
5-S5505 

-87236 
-87265 
-87294 
•87322 
-87351 

6.83443 
6-85218 
6-87001 
6-88792 
6-90592 

-88969 
-88998 

•89027 
.89055 
-89084 

8-06515 
8-08897 
8-11292 
8-13699 
8-16120 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 

.83926 
.83954 
.83983 
-84012 
.  84041 

5.22113 
5-23226 
5.24343 
5.25464 
5.26590 

-85651 
-85680 
-85708 
•85737 
.85766 

5.96900 
5.98301 
5-99708 
6-01120 
6-02538 

-87380 
-87409 
•87438 
•87467 
-87496 

6-92400 
6.94216 
6-96040 
6-97873 
6-99714 

•89113 
•89142 
.89171 
-89200 
.89229 

8-18553 
8-20999 
8-23459 
8-25931 
8-28417 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

-84069 
•84098 
.84127 
.84155 
.84184 

5-27719 
5.28853 
5-29991 
5-31133 
5-32279 

.85795 
.85823 
.85852 
-85881 
-85910 

6-03962 
6-05392 
6-06828 
6-08269 
6.09717 

-87524 
-87553 
-87582 
-87611 
-87640 

7-01565 
7-03423 
7-05291 
7-07167 
7.09052 

-89258 
-89287 
.89316 
.89345 
.89374 

8-30917 
8-33430 
8-35957 
8-38497 
8-41052 

5O 

51 
52 
53 
54 

55 
56 
57 
58 
59 

.84213 
.  84242 
.84270 
•84299 
-84328 

5.33429 
5-34584 
5-35743 
5-36906 
5-38073 

.85939 
-85967 
-85996 
.86025 
.86054 

6.11171 
6.12630 
6.14096 
6.15568 
8-17046 

•87669 
.87698 
-87726 
.87755 
.87784 

7.10946 
7.12849 
7.14760 
7-16681 
7-18612 

-89403 
.89431 
•89460 
.89489 
.89518 

8-43620 
8-46203 
8.48800 
8-51411 
8-54037 

55 
56 
57 
58 

59 

60 

•84357 

5-39245 

.86083 

6.18530 

-87813 

7-20551 

-89547 

8-56677 

60 

249 


TABLE  11.— NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
84°  85°  86° 


/ 

Vers. 

Ex.  sec. 

Vers. 

JSx.  sec. 

Vers. 

Ex.  sec. 

t 

0 

2 
3 
4 

•89547 
•89576 
.89605 
.89634 
.89663 

8.56677 
8.59332 
8.62002 
8-64687 
8.67387 

.91284 
.91313 
.91342 
.91371 
•91400 

10.47371 
10.51199 
10.55052 
10.58932 
10-62837 

.93024 
.93053 
.93082 
.93111 
•93140 

13.33559 
13.39547 
13-45586 
13-51676 
13.57817 

O 
I 

2 
3 
4 
5 
6 
7 
8 
9 

5 
6 
7 
8 
JL 
10 
11 
12 
13 
14 

.89692 
.89721 
.89750 
.89779 
.89808 

8.70103 
8-72833 
8.75579 
8-78341 
8.81119 

.91429 
.91458 
.91487 
-91516 
.91545 

10.66769 
10.70728 
10.74714 
10.78727 
10-82768 

.93169 
.93198 
.93227 
.93257 
•93286 

13-64011 
13-70258 
13-76558 
13-82913 
13.89323 

.89836 
•89865 
.89894 
•89923 
•89952 

8.83912 
8-86722 
8.89547 
8.92389 
8-95248 

•91574 
.91603 
.91632 
.91661 
.91690 

10.86837 
10-90934 
10.95060 
10.99214 
11.03397 

.93315 
.93344 
.93373 
•93402 
•93431 

13-95788 
14-02310 
14-08890 
14-15527 
14-22223 

1O 

11 
12 
13 
14 

15 
16 
17 
18 
2L. 
20 
21 
22 
23 
24 

•89981 
•90010 
•90039 
•90068 
•90097 

8-98123 
9.01015 
9-03923 
9.06849 
9.09792 

•91719 
.91748 
.91777 
.91806 
-91835 

11.07610 
11.11852 
11.16125 
11.20427 
11.24761 

.93460 
.93489 
.93518 
.93547 
-93576 

14-28979 
14-35795 
14-42672 
14-49611 
14-56614 

15 
16 
17 
18 
19 

•90126 
.90155 
•90184 
•90213 
.90242 

9.12752 
9.15730 
9.18725 
9.21739 
9-24770 

•91864 
.91893 
.91922 
•91951 
.91980 

11.29125 
11.33521 
11.37948 
11.42408 
11.46900 

•93605 
-93634 
.93663 
.93692 
.93721 

14-63679 
14-70810 
14-78005 
14-85268 
14.92597 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

25 
26 
27 
28 
29 

•90271 
-90300 
.90329 
.90358 
.90386 

9.27819 
9.30887 
9.33973 
9.37077 
9.40201 

.92009 
.92038 
•92067 
.92096 
-92125 

11.51424 
11.55982 
11.60572 
11.65197 
11.69856 

.93750 
•93779 
.93808 
•93837 
•93866 

14-99995 
15-07462 
15-14999 
15-22607 
15-30287 

30 

31 
32 
33 
34 

•90415 
•90444 
•90473 
•90502 
.90531 

9.43343 
9.46505 
9.49685 
9.52886 
9-56106 

.92154 
.92183 
•92212 
.92241 
.92270 

11.74550 
11.79278 
11.84042 
11.88841 
11-93677 

.93895 
-93924 
.93953 
-93982 
.94011 

15-38041 
15-45869 
15-53772 
15-61751 
15.69808 

35 
36 
37 
38 
39 

.90560 
.90589 
.90618 
.90647 
.90676 

9.59346 
9-62605 
9.65885 
9.69186 
9-72507 

.92299 
.92328 
.92357 
•92386 
.92415 

11.98549 
12.03458 
12.08040 
12.13388 
12.18411 

-  94040 
.94069 
-94098 
.94127 
•94156 

15-77944 
15-86159 
15-94456 
16-02835 
16.11297 

35 
36 
37 
38 

39 

40 

41 
42 
43 
44 

.90705 
.90734 
.90763 
.90792 
.90821 

9.75849 
9.79212 
9.82596 
9.86001 
9-89428 

.92444 
.92473 
.92502 
-92531 
-92560 

12-23472 
12.28572 
12.33712 
12.38891 
12.44112 

.94186 
.94215 
.  94244 
-  94273 
.94302 

16-19843 
16-28476 
16.37196 
16-46005 
16-54903 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

45 
46 
47 
48 
49 

.90850 
.90879 
.90908 
.90937 
.90966 

9-92877 
9.96348 
9.99841 
10-03356 
10.06894 

•92589 
-92618 
.92647 
.92676 
-92705 

12.49373 
12.54676 
12.60021 
12.65408 
12.70838 

-94331 
-94360 
-94389 
-94418 
.  94447 

16-63893 
16-72975 
16-82152 
16-91424 
17-00794 

50 

51 
52 
53 
54 

.90995 
•91024 
.91053 
.91082 
.91111 

10-10455 
10-14039 
10.17646 
10.21277 
10-24932 

.92734 
.92763 
.92792 
.92821 
-92850 

12.76312 
12-81829 
12-87391 
12-92999 
12-98651 

-94476 
.94505 
-94534 
.94563 
.94592 

17.10262 
17.19830 
17.29501 
17-39274 
17-49153 

5O 

51 
52 
53 

54 

55 
56 
57 
58 
59 

.91140 
.91169 
.91197 
.91226 
•91255 

10-28610 
10.32313 
10-36040 
10.39792 
10-43569 

.92879 
.92908 
.92937 
.92966 
.92995 

13.04350 
13.10096 
13.15889 
13.21730 
13-27620 

-94621 
-94650 
.94679 
.94708 
-94737 

17-59139 
17.69233 
17.79438 
17-89755 
18-00185 

55 
56 
57 
58 
59 

60 

.91284 

10.47371 

.93024 

13-33559 

.94766 

18-10732 

60 

250 


TABLE  11. —NATURAL    VERSED    SINES    AND    EXTERNAL    SECANTS. 
87°  88°  89° 


1 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

o 

2 
3 

4 

•94766 
.94795 
.94825 
.94854 
.94883 

18-10732 
18-21397 
18-32182 
18.43088 
18-54119 

.96510 
.96539 
.96568 
•96597 
.96626 

27.65371 
27.89440 
28-13917 
28-38812 
28-64137 

.98255 
.98284 
•98313 
.98342 
-98371 

F.3-  29869 
57.26976 
58-27431 
59-31411 
60.39105 

O 

1 
2 
3 
4 

5 
6 
7 
8 
9 

.94912 
.94941 
.94970 
.94999 
.95028 

18.65275 
18.76560 
18-87976 
18-99524 
19.11208 

.96655 
.96684 
•96714 
.96743 
.96772 

28-89903 
29-16120 
29.42802 
29.69960 
29.97607 

•98400 
.98429 
-98458 
-98487 
-98517 

61-50715 
62.66460 
63-86572 
65-11304 
66-40927 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.95057 
.95086 
.95115 
.95144 
.95173 

19.23028 
19.34989 
19-47093 
19.59341 
19-71737 

.96801 
.96830 
.96859 
.96888 
.96917 

30.25758 
30-54425 
30.83623 
31-13366 
31-43671 

.98546 
•98575 
.98604 
.98633 
-98662 

67-75736 
69-16047 
70-62285 
72-14583 
73-73586 

10 

11 
12 
13 
14 

15 
16 
17 
18 

J9 

.95202 
.95231 
.95260 
.95289 
.95318 

19-84283 
19-96982 
20-09838 
20.22852 
20.36027 

.96946 
.96975 
.97004 
.97033 

•970R2 

31-74554 
32.06030 
32-38118 
32-70835 
33-04199 

-98691 
.98720 
.98749 
.98778 
.98807 

75-39655 
77.13274 
78.94968 
80-85315 

82-84947 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

.95347 
.95377 
.95406 
•95435 
.95464 

20.49368 
20-62876 
20.76555 
20.90409 
21-04440 

.97092 
.97121 
.97150 
.97179 
.97208 

33-38232 
33.72952 
34-08380 
34-44539 
34-81452 

-98836 
-98866 
.98895 
•98924 
.98953 

84-94561 
87-14924 
89-46886 
91-91387 
94-49471 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

.95493 
.95522 
.95551 
.95580 
.95609 

21.18653 
21.33050 
21-47635 
21-62413 
21-77386 

.97237 
.97266 
.97295 
.97324 
.97353 

35-19141 
35-57633 
35-96953 
36-37127 
36.78185 

.98982 
.99011 
.99040 
.99069 
.99098 

97-22303 
100-1119 
103-1757 
106-4311 
109-8966 

25 
26 
27 
28 

29 

30 

31 
32 
33 
34 

.95638 
•95667 
.95696 
.95725 
.95754 

21-92559 
22-07935 
22-23520 
22-39316 
22-55328 

.97382 
.97411 
.97440 
.97470 
-97499 

37.20155 
37-63068 
38-06957 
38.51855 
38-97797 

.99127 
.99156 
.99186 
.99215 
-99244 

113-5930 
117-5444 
121-7780 
126-3253 
131-2223 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

.95783 
.95812 
.95842 
•95871 
.95900 

22-71563 
22-88022 
23-04712 
23-21637 
23-38802 

.97528 
.97557 
•97586 
.97615 
97644 

39-44820 
39-92963 
40-42266 
40-92772 
41-44525 

.99278 
.99302 
•99331 
.99360 
.99389 

136-5111 
142-2406 
148-4684 
155-2623 
162-7033 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.95929 
.95958 
.95987 
.96016 
.96045 

23-56212 
23-73873 
23-91790 
24-09969 
24-28414 

•97673 
.97702 
.97731 
.97760 
.97789 

41-97571 
42-51961 
43-07746 
43-64980 
44-23720 

.99418 
•99447 
.99476 
.99505 
-99535 

170-8883 
179-9350 
189-9868 
201-2212 
213-8600 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 

•96074 
.96103 
.96132 
.96161 
.96190 

24-47134 
24-66132 
24-85417 
25-04994 
25-24869 

.97819 
.97848 
.97877 
.97906 
.97935 

44-84026 
45-45963 
46-09596 
46-74997 
47-42241 

.99564 
.99593 
.99622 
.99651 
.99680 

228-1839 
244-5540 
263-4427 
285-4795 
311-5230 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

.96219 
.96248 
.96277 
.96307 
.96336 

25-45051 
25.65546 
25.86360 
26.07503 
26-28981 

.97964 
.97993 
.98022 
•98051 
-98080 

48.11406 
48-82576 
49-55840 
50.31290 
51-09027 

.99709 
.99738 
•99767 
.99796 
.99825 

342-7752 
380.9723 
428.7187 
490-1070 
571-9581 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

.96365 
.96394 
.96423 
.96452 
.96481 

26-50804 
26-72978 
26-95513 
27-18417 
27-41700 

.98109 
.98138 
.98168 
.98197 
.98226 

51.89156 
52.71790 
53-57046 
54.45053 
55-35946 

.99855 
.99884 
.99913 
.99942 
.99971 

686.5496 
858-4369 
1144-916 
1717-874 
3436-747 

55 
56 
57 
58 
59 

60 

•96510 

27-65371 

.98255 

56-29869 

1.00000 

Infinite 

60 

251 


252     12.  —CUBIC  YARDS  PER  100  FEET.     SLOPES  i  :  1. 


Depth 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

Base 

12 

14 

16 

18 

22 

24 

26 

28 

1 

45 

53 

60 

68 

82 

90  i    97 

105 

2 

93 

107 

122 

137 

167 

181 

196 

211 

3 

142 

163 

186 

208 

253 

275 

297 

319 

4 

193 

222 

252 

281 

341 

370 

400 

430 

5 

245 

282 

319 

356 

431 

468 

605 

542 

6 

300 

344 

389 

433 

522 

567 

611 

656 

7 

356 

408 

460 

512 

616 

668 

719 

771 

8 

415 

474 

533 

593 

711 

770 

830 

889 

9 

475 

542 

608 

675 

808 

875 

942 

1008 

10 

537 

611 

685 

759 

907 

981 

1056 

1130 

11 

601 

682 

764 

845 

1008 

1090 

1171 

1253 

12 

667 

756 

844 

933 

1111 

1200 

1289 

1378 

13 

734 

831 

926 

1023 

1216 

1312 

1408 

1505 

14 

804 

907 

1010 

1115 

1322 

1426 

1530 

1633 

15 

8<3 

986 

1096 

1208 

1431 

1542 

1653 

1764 

16 

948 

1067 

1184 

1304 

1541 

1659 

1778 

1896 

17 

1023 

1149 

1274 

1401 

1653 

1779 

1905 

2031 

18 

1100 

1233 

1366 

1500 

1767 

1900 

2033 

2167 

19 

1179 

1319 

1460 

1601 

1882 

2023 

2164 

2305 

20 

1259 

1407 

1555 

1704 

2000 

2148 

2296 

2444 

21 

1342 

1497 

1653 

1808 

2119 

2275 

2431 

2586 

22 

1426 

1589 

1752 

1915 

2241 

2404 

2567 

*730 

23 

1512 

1682 

1853 

2023 

2364 

2534 

2705 

2875 

24 

1600 

1778 

1955 

2133 

2489 

2667 

2844 

3022 

25 

1690 

1875 

2060 

2245 

2616 

J2801 

2986 

3171 

26 

1781 

1974 

2166 

2359 

2744 

2937 

3130 

3322 

27 

1875 

2075 

2274 

2475 

2875 

3075 

3275 

3475 

28 

1970 

2178 

2384 

2593 

3007 

3215 

8422 

3630 

29 

2068 

2282 

2496 

2712 

3142 

3356 

3571 

3786 

30 

2167 

2389 

2610 

2833 

3278 

3500 

3722 

3944 

31 

2288 

2497 

2726 

2956 

3416 

3645 

3875 

4105 

32 

2370 

2607 

2844 

3081 

3556 

3793 

4030 

4267 

33 

2475 

2719 

2964 

3208 

3697 

3942 

4186 

4431 

34 

2581 

2833 

3085 

3337 

3841 

4093 

4344 

4596 

35 

2690 

2949 

3208 

3468 

3986 

4245 

4505 

4764 

36 

2800 

3067 

3333 

3600 

4133 

4400 

4667 

4933 

37 

2912 

3186 

3460 

3734 

4282 

4556 

4831 

5105 

38 

3026 

3307 

3589 

3870 

4433 

4715 

4996 

5278 

39 

3142 

3431 

3719 

4008 

4586 

4875 

6164 

5453 

40 

3259 

3556 

3852 

4148 

4741 

5037 

5333 

6630 

41 

3379 

3682 

3986 

4290 

4897 

5201 

5505 

5808 

42 

3500 

3811 

4122 

4433 

5056 

5367 

6678 

5989 

43 

3623 

3942 

4260 

4579 

5216 

5534 

5853 

6171 

44 

3748 

4074 

4400 

4726 

5378 

5704 

6030 

6356 

45 

3875 

4208 

4541 

4875 

5542 

5875 

6208 

6542 

46 

4004 

4344 

4684 

5026 

5707 

6048 

6389 

6730 

47 

4134 

4482 

4830 

5179 

5875 

6223 

6571 

6919 

48 

4267 

4622 

4978 

5333 

6044 

6400 

6756 

7111 

49 

4401 

4764 

5127 

5490    6216 

6579 

6943 

7305 

50 

4537 

4907 

5278 

5648 

6389 

6759 

7130 

7500 

51 

4675 

5053 

5430 

5808 

6564 

6942 

7319 

7697 

52 

4815 

5200 

5584 

597'0 

6741 

7126 

7511 

7896 

53 

4956 

5349 

5741 

6134  i  6919 

7312 

7705 

8097 

54 

5100 

5500 

5900 

6300  !  7100 

7500 

7900 

&300 

55 

5245 

5653 

6060 

6468 

7282 

7690 

8097 

8505 

56 

5393 

5807 

6222 

6637 

7467 

7881 

8296 

8711 

57 

5542 

5964 

6386 

6808 

7653 

8075 

8497 

8919 

58 

5693 

6122 

6552 

6981 

7841 

8270 

8700 

9130 

59 

5845 

6282 

6719 

7156 

8031 

3468 

8905 

9342 

60 

6000 

6444 

6889 

7333 

8222 

8667 

9111 

9556 

12.— CUBIC  YARDS  PER  100  FEET.     SLOPES  *  :  1.  253 


Depth 

Base 

12 

Base 
14 

Base 
1G 

Base 
13 

Base 
22 

Base 
24 

Base 
26 

Base 
28 

1 

46 

54 

Cl 

CD 

83 

91 

98 

106 

2 

96 

111 

126 

141 

170 

185 

200 

215 

3 

150 

172 

194 

217 

261 

283 

306 

328 

4 

207 

237 

267 

296 

356 

385 

415 

444 

5 

269 

306 

343 

380 

454 

491 

528 

565 

6 

333 

378 

422 

467 

556 

600 

644 

689 

7 

402 

454 

506 

557 

661 

713 

765 

817 

8 

474 

533 

593 

652 

770 

830 

889 

948 

9 

550 

617 

683 

750 

883 

950 

1017 

1083 

10 

630 

704 

778 

852 

1000 

1074 

1148 

1222 

11 

713 

794 

876 

957 

1120 

1202  * 

1283 

1365 

12 

800 

889 

978 

1067 

1244 

1333 

1422 

1511 

13 

891 

987 

1083 

1180 

1372 

1469 

1565 

1661 

14 

985 

1089 

1193 

1296 

1504 

1607 

1711 

1815 

15 

1083 

1194 

1306 

1417 

1639 

1750 

1861 

1972 

16 

1185 

1304 

1422 

1541 

1779 

1896 

2015 

2133 

17 

1291 

1417 

1543 

1669 

1920 

2046 

2172 

2298 

18 

1400 

1533 

1667 

1800 

2067 

2200 

2333 

2467 

19 

1513 

1654 

1794 

1935 

2217 

2357 

2498 

2639 

20 

1630 

1778 

1926 

2074  ' 

2370 

2519 

2667 

2815 

21 

1750 

1906 

2061 

2217 

2528 

2683 

2839 

2994 

22 

1874 

2037 

2200 

2363 

2689 

2852 

3015 

3178 

23 

2002 

2172 

2343 

2513 

2854 

3024 

3194 

8365 

24 

2133 

2311 

2489 

2667 

3022 

3200 

3378 

3556 

25 

2269 

2454 

2639 

2824 

3194 

3380 

3565 

3750 

26 

2407 

2600 

2793 

2985 

3370 

35C3 

3756 

3948 

27 

2550 

2750 

2950 

3150 

3550 

3750 

3950 

4151 

28 

2696 

2904 

3111 

3319 

3733 

3941 

4148 

4356 

29 

2846 

3061 

3276 

3491 

3920 

4135 

4350 

4565 

30 

3000 

3222 

3444 

3667 

4111 

4333 

4556 

4778 

31 

3157 

3387 

3617 

3846 

4306 

4535 

4765 

4994 

32 

3319 

3556 

3793 

4030 

4504 

4741 

4978 

5215 

33 

3483 

3728 

3972 

4217 

4706 

4950 

5194 

5439 

34 

3652 

3904 

4156 

4407 

4911 

5163 

5415 

5667 

35 

3824 

4083 

4343 

4602 

5120 

5380 

5639 

5898 

36 

4000 

4267 

4533 

4800 

5333 

5600 

5867 

6133  . 

37 

4180 

4454 

4728 

5002 

5550 

5824 

6098 

6372 

38 

4363 

4644 

4926 

5207 

5770 

6052 

6333 

6615 

39 

4550 

4839 

5128 

5417 

5994 

6283 

6572 

6861 

40 

4741 

5037 

5333 

5630 

6222 

6519 

6815 

7111 

41 

4935 

5239 

5543 

5846 

6454 

6757 

7061 

7365 

42 

5133 

5444 

5756 

6067 

6689 

7000 

7311 

7623 

43 

5335 

5654 

5972 

6291 

6928 

7246 

7565 

7883 

44 

5541 

5867 

6193 

6519 

7170 

7496 

7822 

8148 

45 

5750 

60&3 

6417 

6750 

7417 

7750 

8083 

8417 

46 

5963 

6304 

6644 

6985 

7667 

8007 

8348 

8689 

47 

6180 

6528 

6876 

7224 

7920 

8269 

8617 

8965 

48 
49 

6400 
6624 

6756 
6987 

7111 
7350 

m 

m 

8889 
9165 

50 

6852 

7222 

7593 

7963 

87C4 

9074 

9444 

9815 

51 

7083 

7461 

7839 

8217 

8972 

9350 

9728 

10106 

52. 

7319 

7704 

8089 

8474 

9244 

9630 

10015 

10400 

53 

7557 

7950 

8343 

8735 

9520 

9913 

10306 

10698 

54 

7800 

8200 

8600 

9000 

9800 

10200 

10600 

11000 

55 

8046 

8454 

8861 

9269 

10083 

10491 

10898 

11306 

56 

8296 

8711 

9126 

9541 

10370 

10785 

11200 

11615 

57 

8550 

8972 

9394 

9817 

10661 

11083 

11506 

11928 

58 

8807 

9237 

9667 

10096 

10956 

11385 

11815 

12244 

59 

9069 

9506 

9943 

10380 

11254 

11691 

12128 

12565 

60 

9333 

9778 

10222 

10667 

11556 

12000 

12444 

12889 

254      12.— CUBIC  YARDS  PER  100  FEET.     SLOPES  1  :  1. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

1 

48 

56 

63 

70 

78 

107 

115 

122 

• 

104 

119 

133 

148 

163 

222 

237 

252 

3 

167 

189 

211 

233 

256 

344 

367 

389 

237 

267 

296 

326 

356 

474 

504 

533 

5 

315 

352 

389 

426 

463 

611 

648 

685 

6 

400 

444 

489 

533 

578 

756 

800 

844 

7 

493 

544 

596 

648 

700 

907 

959 

1011 

8 

693 

652 

711 

77'0 

830 

1067 

1126 

1185 

9 

700 

767 

833 

900 

967 

1233 

1300 

1367 

10 

815 

889 

963 

1037 

1111 

1407 

1481 

1556 

11 

937 

1019 

1100 

1181 

1263 

1589 

1670 

1752 

12 

1067 

1156 

1244 

1333 

1422 

1778 

1867 

1956 

13 

1204 

1300 

1396 

1493 

1589 

1974 

2070 

2167 

14 

1348 

1452 

1556 

1659 

1763 

2178 

2281 

2385 

15 

1500 

1611 

1722 

1833 

1944 

2389 

2500 

2611 

16 

1659 

1778 

1896 

2015 

2133 

2607 

2726 

2844 

17 

1826 

1952 

2078 

2204 

2330 

2833 

2959 

3085 

18 

2000 

2133 

2267 

2400 

2533 

3067 

3200 

3333 

19 

2181 

2322 

2463 

2604 

2744 

3307 

3448 

3589 

20 

2370 

2519 

2667 

2815 

2963 

3556 

3704 

3852 

21 

2567 

27£2 

2878 

3033 

3189 

3811 

3967 

4122 

22 

2770 

2933 

3096 

3259 

3422 

4074 

4237 

4444 

23 

2981 

8152 

3322 

3493 

3663 

4344 

4515 

4685 

24 

3200 

3378 

3556 

3733 

3911 

4622 

4800 

4978 

25 

3426 

3611 

3796 

3981 

4167 

4907 

5093 

5278 

26 

8659 

3852 

4044 

4237 

4430 

5200 

5393 

5585 

27 

3900 

4100 

4300 

4500 

4700 

5500 

5700 

5900 

28 

4148 

4356 

4563 

4770 

4978 

5807 

6015 

6222 

29 

4404 

4619 

4833 

5048 

5263 

6122 

6337 

6552 

30 

4667 

4889 

5111 

5333 

5556 

6444 

6667 

6889 

31 

4937 

5167 

5396 

5626 

5856 

6774 

7004 

7233 

32 

5215 

5452 

5689 

5926 

6163 

7111 

7348 

7585 

33 

5500 

5744 

5989 

6233 

6478 

7456 

7700 

7944 

34 

6793 

6044 

6296 

6548 

6800 

7807 

8059 

8311 

35 

6093 

6352 

6611 

6870 

7130 

8167 

8426 

8685 

,  36 

6400 

6667 

6933 

7200 

7467 

8533 

8800 

9067 

37 

6715 

6989 

7263 

7537 

7811 

8907 

9181 

9456 

38 

7037 

7319 

7600 

7881 

8163 

9289 

9570 

9852 

39 

7367 

7656 

7944 

8233 

8522 

9678 

9967 

10256 

40 

7704 

8000 

8296 

8593 

8889 

10074 

10370 

10667 

41 

8048 

8352 

8656 

8959 

9263 

10478 

10781 

11085 

42 

8400 

8711 

9022 

9333 

9644 

10889 

11200 

11511 

43 

8759 

9078 

9396 

9715 

10033 

11307 

11626 

11944 

44 

9126 

9452 

9778 

10104 

10430 

11733 

12059 

12385 

45 

9500 

9833 

10167 

10500 

10833 

12167 

12500 

12833 

46 

9881 

10222 

10563 

10904 

11244 

12607 

12948 

13289 

47 

10270 

10619 

10967 

11315 

11663 

13056 

13404 

13752 

48 

10667 

11022 

11378 

11733 

12089 

13511 

13867 

14222 

49 

11070 

11433 

11796 

12159 

12522 

13974 

14337 

14700 

60 

11481 

11852 

12222 

12593 

12963 

14444 

14815 

15185 

51 

11900 

12278 

12656 

13033 

13411 

14922 

15300 

15678 

62 

12326 

12711 

13096 

13481 

13867 

15407 

15793 

16178 

63 

12759 

13152 

13544 

13937 

14330 

15900 

16293 

16G85 

64 

13200 

13600 

14000 

14400 

14800 

16400 

16800 

17200 

65 

13648 

14056 

14463 

14870 

15278 

16907 

17315 

17722 

66 

14104 

14519 

14933 

15348 

15763 

17422 

17837 

18252 

67 

14567 

14989 

15411 

15833 

16256 

17944 

18367 

18789 

68 

15037 

15467 

15896 

16326 

16756 

18474 

18904 

19333 

69 

15515 

15952 

16389 

16826 

17263 

19011 

19448 

19885 

60 

16000 

16444 

16889 

17333 

17778 

19556 

20000 

20444 

12.— CUBIC  YARDS  PER  100  FEET.     SLOPES  H  :  1.  255 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

1 

50 

57 

65 

72 

80 

109 

117 

124 

2 

111 

126 

141 

156 

170 

230 

244 

259 

3 

183 

206 

228 

250 

272 

361 

383 

406 

4 

267 

296 

326 

356 

385 

504 

533 

563 

6 

361 

398 

435 

472 

509 

657 

694 

731 

6 

467 

511 

556 

600 

644 

822 

867 

911 

7 

583 

635 

687 

739 

791 

998 

1050 

1102 

8 

71i 

770 

830 

889 

948 

1185 

1244 

1304 

9 

850 

917 

983 

1050 

1116 

1383 

1450 

1517 

10 

1000 

1074 

1148 

1222 

1296 

1593  * 

1667 

1741 

11 

1161 

1243 

1324 

1406 

1487 

1813 

1894 

1976 

12 

1333 

1422 

1511 

1600 

1689 

2044 

2133 

2222 

13 

1517 

1613 

1709 

1806 

1902 

2287 

2383 

2480 

14 

1711 

1815 

1919 

2022 

2126 

2541 

2644 

2748 

15 

1917 

2028 

2139 

2250 

2361 

2806 

2917 

3028 

16 

2133 

2252 

2370 

2489 

2607 

3081 

3200 

3319 

1? 

2361 

2487 

2613 

2739 

2865 

3369 

3494 

3620 

18 

2600 

2733 

2867 

3000 

3133 

3667 

3800 

3933 

19 

2850 

2991 

3131 

3272 

3413 

3976 

4117 

4257 

20 

3111 

3259 

3407 

3556 

3704 

4296 

4444 

4592 

21 

3383 

3539 

3694 

3850 

4005 

4628 

4783 

4939 

22 

3667 

3830 

3993 

4156 

4318 

4970 

5133 

5296 

23 

3961 

4131 

4302 

4472 

4642 

5324 

5494 

5665 

24 

4267 

4444 

4622 

4800 

4978 

5689 

5867 

6044 

25 

4583 

4769 

4954 

5139 

5324 

6065 

6250 

6435 

26 

4911 

5104 

5296 

5489 

5681 

6452 

6644 

6837 

2? 

5250 

5450 

5650 

5850 

6050 

6850 

7050 

7250 

28 

5600 

5807 

6015 

6222 

6430 

7259 

7467 

7674 

2ft 

5961 

6176 

6391 

6606 

6820 

7680 

7894 

8109 

30 

6333 

6556 

6778 

7000 

7222 

8111 

8333 

8555 

31 

6717 

6946 

7176 

7406 

7635 

8554 

8783 

9013 

32 

7111 

7348 

7585 

7822 

8059 

9007 

9244 

9482 

33 

7517 

7761 

8006 

8250 

8494 

9472 

9717 

9962 

34 

7933 

8185 

8437 

8689 

8941 

9948 

10200 

10452 

35 

8361 

8620 

8880 

9139 

9398 

10435 

10694 

10954 

36 

8800 

9067 

9333 

9600 

9867 

10933 

11200 

11467 

37 

9250 

9524 

9798 

10072 

10346 

11443 

11717 

11991 

38 

9711 

9993 

10274 

10556 

10837 

11963 

12244 

12526 

39 

10183 

10472 

10761 

11050 

11339 

12494 

12783 

13072 

40 

10667 

10963 

11259 

11556 

11852 

13037 

13333 

13630 

41 

11161 

11465 

11769 

12072 

12376 

13591 

13894 

14198 

42 

11667 

11978 

12289 

12600 

12911 

14156 

14467 

14778 

43 

12183 

12502 

12820 

13139 

13457 

14731 

15050 

15369 

44 

12711 

13037 

13363 

13689 

14015 

15319 

15644 

15970 

45 

13250 

13583 

13917 

14250 

14583 

15917 

16250 

16583 

46 

13800 

14141 

14481 

14822 

15163 

16526 

16867 

17207 

47 

14361 

14709 

15057 

15406 

15754 

17146 

17494 

17843 

48 

14933 

15289 

15644 

16000 

16356 

17778 

18133 

18489 

49 

15517 

15880 

16243 

16606 

16968 

18420 

18783 

19146 

50  . 

16111 

16481 

16852 

17222 

17592 

19074 

19444 

19815 

51 

16717 

17094 

17472 

17850 

18228 

19739 

20117 

20494 

52 

17333 

17719 

18104 

18489 

18874 

20415 

20800 

21185 

53 

17961 

18354 

18746 

19139 

19531 

21102 

21494 

21887 

54 

18600 

19000 

19400 

19800 

20200 

21800 

22200 

22600 

55 

19250 

19657 

20065 

20472 

20880 

22509 

22917 

23324 

56 

19911 

20326 

20741 

21156 

21570 

23230 

23644 

24059 

57 

20583  , 

21006 

21428 

21850 

22272 

23961 

24383 

24805 

58 

21267 

21696 

22126 

22556 

22985 

24704 

25133 

25563 

59 

21961 

22398 

22835 

23272 

23709 

25457 

25894 

26332 

60 

22667 

23111 

23556 

24000 

24444 

26222 

26667 

27111 

256      12.— CUBIC  YARDS  PER  100  FEET.     SLOPES  2  :  1. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Ba«e 
28 

Base 
30 

Base 
32 

1 

52 

59 

67 

74 

81 

111 

119 

126 

2 

119 

133 

143 

1C3 

178 

237 

252 

267 

3 

200 

222 

244 

267 

289 

878 

400 

422 

4 

296 

326 

856 

385 

415 

5-33 

563 

693 

5 

407 

444 

481 

519 

556 

V04 

741 

778 

6 

533 

578 

622 

637 

711 

889 

933 

979 

7 

674 

726 

778 

830 

881 

1089 

1141 

1193 

8 

830 

889 

948 

1007 

1067 

1304 

1363 

1422 

9 

1000 

1067 

1133 

1200 

1267 

1533 

1600 

1667 

10 

1185 

1259 

1333 

1407 

1481 

1778 

1852 

1926 

11 

1385 

1467 

1548 

1630 

1711 

2037 

2119 

2200 

12 

1600 

1689 

1778 

1867 

1956 

2311 

2400 

2489 

13 

1830 

1926 

2022 

2119 

2215 

2600 

2696 

2793 

14 

2074 

2178 

2281 

2385 

2489 

2904 

3007 

3111 

15 

2333 

2444 

2356 

2667 

2778 

3222 

3333 

3444 

16 

2607 

2726 

2844 

2963 

3081 

&55(3 

3674 

3793 

17 

2896 

3022 

8148 

3274 

3400 

3904 

4030 

4156 

18 

3200 

8333 

3457 

3600 

3733 

4207 

4400 

4533 

19 

3519 

3059 

3800 

3941 

4081 

4644 

4785 

4926 

20 

3852 

4000 

4148 

4296 

4444 

5037 

5185 

5333 

21 

4200 

4356 

4511 

4667 

4822 

5444 

5600 

5756 

22 

4563 

4730 

4889 

5053 

5215 

5867 

6030 

6193 

23 

4941 

5111 

5281 

5452 

5622 

6304 

6474 

6644 

24 

5333 

5511 

5689 

5867 

6044 

6756 

6933 

7111 

25 

5741 

5926 

6111 

6296 

6481 

7222 

7407 

7593 

26 

6163 

6356 

6548 

6741 

6933 

7704 

7896 

8089 

27 

6600 

6800 

7000 

7200 

7400 

8200 

8400 

8600 

28 

7052 

7259 

7467 

7674 

7881 

8711 

8919 

9126 

29 

7519 

7733 

7948 

8163 

8378 

9237 

9452 

9667 

30 

8000 

8222 

8444 

8667 

8889 

9778 

10000 

10222 

31 

8496 

8726 

8956 

9185 

9415 

10333 

10563 

10793 

32 

9007 

9244 

9481 

9719 

9956 

10904 

11141 

11378 

33 

9533 

9778 

10022 

10267 

10511 

11489 

11733 

11978 

34 

10074 

10326 

10378 

10830 

11081 

12089 

12341 

12593 

35 

10630 

10889 

11148 

11407 

11667 

12704 

12963 

13222 

36 

11200 

11467 

11733 

12000 

12267 

13333 

13600 

13867 

37 

11785 

12059 

12333 

12607 

12881 

13978 

14252 

14526 

38 

12385 

12667 

12948 

13230 

13511 

14637 

14919 

15200 

39 

13000 

13289 

13578 

13867 

14156 

15311 

15GOO 

15889 

40 

13630 

13926 

14223 

14519 

14815 

16000 

16296 

16593 

41 

14274 

14578 

14881 

15185 

15489 

16704 

17007 

17311 

42 

14'.)33 

15244 

15556 

15867 

16178 

17422 

17733 

18044 

43 

15607 

15926 

16224 

16563 

16881 

18156 

18474 

18793 

44 

16296 

16622 

16948 

17274 

17600 

16904 

19230 

19556 

45 

17000 

17333 

17667 

18000 

18333 

19667 

20000 

20333 

46 

17719 

18059 

18400 

18741 

19081 

20444 

20785 

21126 

47 

18452 

18800 

19148 

19496 

19844 

21237 

21585 

21933 

48 

19200 

19556 

19911 

20267 

20622 

22044 

22400 

22756 

49 

19963 

20326 

20689 

21052 

21415 

22867 

23230 

23593 

50 

20741 

20711 

21481 

21852 

22222 

23704 

24074 

24444 

51 

21533 

21911 

22289 

22667 

23044 

24556 

24933 

25311 

*2 

22341 

22726 

23111 

23496 

23881 

25422 

25807 

26193 

53 

23163 

2J556 

23948 

24341 

24733 

26304 

26696 

2?089 

54 

24000 

21400 

24800 

25200 

25600 

27200 

27600 

28000 

55 

24852 

25259 

25667 

26074 

26481 

28111 

28519 

28926 

56 

25719 

26133 

26548 

26963 

27378 

29037 

29452 

29867 

57 

26600 

27022 

27444 

27867 

28289 

20978 

30400 

30822- 

58 

27496 

27926 

28356 

28785 

29215 

30933 

31363 

31793 

59 

28407 

28844 

29281 

29719 

30156 

31904 

32341 

32778 

60 

89333 

29778 

30222 

30667 

31111 

32889 

33333 

33778 

12.— CUBIC  YARDS  PER  100  FEET.    SLOPES  3  :  1.  257 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

1 

56 

63 

70 

78 

85 

115 

122 

130 

2 

133 

148 

163 

178 

193 

252 

267 

281 

3 

233 

256 

278 

300 

322 

411 

433 

456 

4 

356 

385 

415 

444 

474 

593 

622 

652 

5 

500 

537 

574 

611 

648 

796 

833 

870 

6 

667 

711 

756 

800 

&44 

1022 

1067 

1111 

7 

856 

907 

959 

1011 

1063 

1270 

1322 

1374 

8 

1067 

1126 

1185 

1244 

1304 

1541 

1600 

1659 

9 

1300 

1367 

1433 

1500 

1567 

1833 

1900 

1967 

10 

1556 

1630 

1704 

1778 

1852 

2148 

2222 

2296 

11 

1833 

1915 

1996 

2078 

2159 

2485 

2567 

2648 

IS 

2133 

2222 

2311 

2400 

2489 

2844 

2933 

3022 

13 

2456 

2552 

2648 

2744 

2841 

3226 

3322 

3419 

14 

2800 

2904 

3007 

3111 

3215 

3630 

3733 

3837 

15 

3167 

3278 

3389 

3500 

3611 

4056 

4167 

4278 

16 

3556 

3674 

3793 

3911 

4030 

4504 

4622 

4741 

17 

3967 

4093 

4219 

4344 

4470 

4974 

5100 

5226 

18 

4400 

4533 

4667 

4800 

4933 

5467 

5600 

5733 

19 

4856 

4996 

5137 

5278 

5419 

5981 

6122 

6263 

20 

5333 

5481 

5630 

5778 

5926 

6519 

6667 

6815 

21 

5833 

5989 

6144 

6300 

6456 

7078 

7233 

7389 

22 

6356 

6519 

6681 

6844 

7007 

7659 

7822 

7985 

23 

6900 

7070 

7241 

7411 

7581 

8263 

8433 

8504 

24 

7467 

7644 

7822 

8000 

8178 

8889 

9067 

9144 

25 

8056 

8241 

8426 

8611 

8796 

9537 

9722 

9807 

26 

8667 

8859 

9052 

9244 

9437 

10207 

10400 

10593 

27 

9300 

9500 

9700 

9900 

10100 

10900 

11100 

11300 

28 

9956 

10163 

10370 

10578 

10785 

11615 

11822 

12030 

29 

10633 

10848 

11063 

11278 

11493 

12352 

12567 

12781 

30 

11333 

11556 

11778 

12000 

12222 

13111 

13333 

13556 

31 

12056 

12285 

12515 

12744 

12974 

13893 

14122 

14352 

32 

12800 

13037 

13274 

13511 

13748 

14696 

14933 

15170 

as 

13567 

13811 

14056 

14300 

14544 

15522 

15767 

16011 

34 

14356 

14607 

14859 

15111 

15363 

16370 

16622 

16874 

35 

15167 

15426 

15685 

15944 

16204 

17241 

17500 

17759 

36 

16000 

16267 

16533 

1G800 

17067 

18133 

18400 

18667 

37 

16856 

17130 

17404 

17678 

17959 

19048 

19322 

19596 

38 

17733 

18015 

18296 

18578 

18859 

19985 

20267 

20548 

89 

18633 

18922 

19211 

19500 

19789 

20944 

21233 

21522 

40 

19556 

19852 

20148 

20444 

20741 

21926 

22222 

22516 

41 

20500 

20804 

21107 

21411 

21715 

22930 

23233 

23537 

42 

21467 

21778 

22089 

22400 

22711 

23956 

24267 

24578 

43 

22456 

22774 

23093 

23411 

23730 

25004 

25322 

25641 

44 

23467 

23793 

24119 

24444 

24770 

26074 

26400 

26726 

45 

24500 

24833 

25167 

25500 

25833 

27167 

27500 

27833 

46 

25556 

25896 

26237 

26578 

26919 

28281 

28622 

28963 

47 

26633 

26981 

27330 

27678 

28026 

29419 

29767 

30115 

48 

27733 

28089 

28800 

29156 

30578 

30938 

31289 

49 

28856 

20219 

21)581 

29944 

30307 

31759 

32122 

32485 

50 

30000 

30370 

30741 

31111 

31481 

32963 

33333 

33704 

51 

31167 

31544 

31922 

32300 

32678 

34189 

34567 

34944 

52 

32356 

32741 

33126 

33511 

33396 

35437 

35822 

36207 

53 

33567 

33959 

34352 

34744 

35137 

36707 

37100 

87493 

54 

34800 

35200 

35600 

36000 

36400 

38000 

38400 

38800 

55 

36056 

36463 

36870 

37278 

37685 

39315 

39722 

4€130 

56 

37333 

37748 

38163 

38578 

38993 

40652 

41067 

41481 

5? 

38633 

39056 

39478 

39900 

40322 

42011 

42433 

42856 

58 

39956 

40385 

40815 

41244 

41674 

43393 

43822 

44252 

59 

41300 

41737 

42174 

42611 

43048 

44796 

45233 

45670 

60 

42667 

43111 

43556 

44000 

44444 

46222 

46667 

47111 

258  TABLE  13.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 

IS: 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

Area. 
Sq, 
Ft. 

Cubic 
Yards. 

Area. 
Sq. 
Ft. 

Cubic 
Yards. 

1 

3.7 

51 

188.9 

101 

374.1 

151 

559.3 

201 

744.4 

2 

7.4 

52 

192.6 

102 

377.8 

152 

563.0 

202 

748.2 

3 

11.1 

53 

196.3 

103 

381.5 

153 

566.7 

203 

751.9 

4 

14.8 

54 

200.0 

104 

385.2 

154 

570.4 

204 

755.6 

5 

18.5 

55 

203.7 

105 

388.9 

155 

574.1 

905 

759.3 

6 

22.2 

56 

207.4 

106 

392.6 

156 

577.8 

206 

763.0 

7 

25.9 

57 

211.1 

107 

396.3 

157 

581.5 

207. 

766.7 

8 

29.6 

58 

214.8 

108 

400.0 

158 

585.2 

208 

770.4 

9 

33.3 

59 

218.5 

109 

403.7 

159 

588.9 

209 

774.1 

10 

37.0 

60 

222.2 

110 

407.4 

160 

592.6 

210 

777.8 

11 

40.7 

61 

225.9 

111 

411.1 

161 

596.3 

211 

781  5 

12 

44.4 

62 

229.6 

112 

414.8 

162 

600.0 

212 

785.2 

13 

48.1 

63 

233.3 

113 

418.5 

163 

603.7 

213 

788.9 

14 

51.9 

64 

237.0 

114 

422.2 

164 

607.4 

214 

792.6 

15 

55.6 

65 

240.7 

115 

425.9 

165 

611.1 

215 

796.3 

16 

59.3 

66 

244  .4 

116 

429.6 

166 

614.8 

216 

800.0 

17 

63.0 

67 

248.2 

117 

433.3 

167 

618.5 

217 

803.7 

18 

66.7 

68 

251.9 

118 

437.0 

168 

622.2 

218 

807.4 

19 

70.4 

69 

255.6 

119 

440.7 

169 

625.9 

219 

811.1 

20 

74.1 

70 

259.3 

120 

444.4 

170 

629.6 

220 

814.8 

21 

77.8 

71 

263.0 

121 

448.2 

171 

633.3 

221 

818.5 

22 

81.5 

72 

266.7 

123 

451.9 

172 

637.0 

222 

822.2 

23 

85.2 

73 

270.4 

123 

455  6 

173 

640.7 

223 

825.9 

24 

88.9 

74 

274.1 

124 

459.3 

174 

644.4 

224 

829.6 

25 

92.6 

75 

277.8 

125 

463.0 

175 

648.2 

225 

833.3 

26 

96.3 

76 

281.5 

126 

466.7 

176 

651.9 

226 

837.0 

27 

100.0 

77 

285  2 

127 

470.4 

177 

655.6 

227 

840.7 

28 

103.7 

78 

,  288.9 

128 

474  1 

178 

659.3 

228 

844.4 

29 

107.4 

79 

292.6 

129 

477.8 

179 

663.0 

229 

848.2 

30 

111.1 

80 

296.3 

130 

481.5 

180 

666.7 

230 

851.9 

31 

114.8 

81 

300.0 

131 

485.2 

181 

670.4 

231 

855.6 

32 

118.5 

82 

303.7 

132 

488  9 

182 

674.1 

232 

859.3 

33 

122.2 

83 

307.4 

133 

492.6 

183 

677.8 

233 

863.0 

34 

125.9 

84 

311.1 

134 

496.3 

184 

681.5 

234 

866.7 

35 

129  6 

85 

314.8 

135 

500.0 

185 

685.2 

235 

870.4 

36 

133.3 

86 

318.5 

136 

503.7 

186 

688.9 

236 

874.1 

37 

137.0 

87 

322.2 

137 

507.4 

187 

692.6 

237 

877.8 

38 

140.7 

88 

325.9 

138 

511.1 

188 

696.3 

238 

881.5 

39 

144  4 

89 

329.6 

139 

514.8 

189 

700.0 

235 

885.2 

40 

148.2 

90 

333.3 

140 

518.5 

190 

703.7 

240 

888.9 

41 

151.9 

91 

337.0 

141 

522.2 

191 

707.4 

241 

892.6 

42 

155.6 

92 

340.7 

142 

525.9 

192 

711.1 

242 

896.3 

43 

159  3 

93 

344.4 

143 

529.6 

193 

714.8 

243 

900.0 

44 

163.0 

94 

348.2 

144 

533.3 

194 

718.5 

244 

903.7 

45 

166.7 

95 

351.9 

145 

537.0 

195 

722.2 

245 

907.4 

46 

170.4 

96 

355.6 

146 

540.7 

196 

725.9 

246 

911.1 

47 

174.1 

97 

359.3 

147 

544.4 

197 

729.6 

247 

914.8 

48 

177.8 

96 

363.0 

148 

548.2 

198 

733.3 

248 

918.5 

49 

181.5 

99 

366.7 

149 

551.9 

199 

737.0 

249 

922.2 

50 

185.2 

100 

370.4 

150 

555.6 

200 

740.7 

250 

925.9 

TABLE  13.— CUBIC  YARDS  IN  100  FEET  LENGTH.  259 


Area 

* 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

Area 

ft 

Cubic 
Yards. 

Area 

•ft 

Cubic 
Yards. 

Area 

ft 

Cubic 
Yards. 

251 

929.6 

301 

1114.8 

351 

1300.0 

401 

1485.2 

451 

1670.4 

252 

933.3 

302 

1118.5 

352 

1303.7 

402 

1488.9 

152 

1674.1 

253 

93?.  0 

303 

1122.2 

353 

1307.4 

403 

1492.6 

453 

1677.8 

254 

940.7 

304 

1125.9 

a54 

1311.1 

404 

1496.3 

454 

1681.5 

255 

944.4 

305 

1129.6 

355 

1314.8 

405 

1500.0 

455 

1685.2 

256 

948.2 

306 

1133.3 

356 

1318.5 

406 

1503.7 

456 

1688.9 

1      257 

951.9 

307 

1137.0 

357 

1322.2 

407 

1507.4 

457 

1692.6 

258 

955.6 

308 

1140.7 

358 

1325.9 

408 

1511.1 

458 

1696.3 

259 

959.3 

309 

1144.4 

359 

1329.6 

409 

1514.8 

459 

1700.0 

260 

963.0 

310 

1148.2 

360 

1333.3 

410 

1518.5 

460 

1703.7 

261 

966.7 

311 

1151.9 

361 

1337.0 

411 

1522.2. 

461 

1707.4 

262 

970.4 

312 

1155.6 

362 

1340.7 

412 

1525.9 

462 

1711.1 

263 

974.1 

313 

1159.3 

363 

1344.4 

413 

1529.6 

463 

1714.8 

264 

977.8 

314 

1163.0 

364 

1348.2 

414 

1533.3 

-164 

1718.5 

265 

981.5 

315 

1166.7 

365 

1351.9 

415 

1537.0 

465 

1722.2 

266 

985.2 

316 

1170.4 

366 

1355.6 

416 

1540.7 

466 

1725.9 

267 

988.9 

317 

1174.1 

367 

1359.3 

417 

1544.4 

467 

1729.6 

268 

992.6 

318 

1177.8 

368 

1363.0 

418 

1548.2 

468 

1733.3 

269 

996.3 

319 

1181.5 

369 

1366.7 

419 

1551.9 

469 

1737.0 

270 

1000.0 

320 

1185.2 

370 

1370.4 

420 

1555.6 

470 

1740.7 

271 

1003.7 

321 

1188.9 

371 

1374.1 

421 

1559.3 

471 

1744.4 

272 

1007.4 

322 

1192.6 

372 

1377.8 

422 

1563.0 

472 

1748.2 

273 

1011.1 

323 

1196.3 

373 

1381.5 

423 

1566.7 

473 

1751.9 

274 

1014.8 

324 

1200.0 

374 

1385.2 

424 

1570.4 

474 

1755.6 

275 

1018.5 

325 

1203.7 

375 

1388.9 

425 

1574.1 

475 

1759.3 

276 

1022.2 

326 

1207.4 

376 

1392.6 

426 

1577.8 

476 

1763.0 

277 

1025.9 

327 

1211.1 

377 

1396.3 

427 

1581.5 

477 

1766.7 

278 

1029.6 

328 

1214.8 

378 

1400.0 

428 

1585.2 

478 

1770.4 

279 

1033.3 

329 

1218.5 

379 

1403.7 

429 

1588.9 

479 

1774.1 

280 

1037.0 

330 

1222.2 

380 

1407.4 

430 

1592.6 

480 

1777.8 

281 

1040.7 

331 

1225.9 

381 

1411.1 

431 

1596.3 

481 

1781.5 

282 

1044.4 

332 

1229  6 

382 

1414.8 

432 

1600.0 

482 

1785.2 

283 

1048.2 

333 

1233.3 

383 

1418.5 

433 

1603.7 

483 

1788.9 

284 

1051.9 

334 

1237.0 

384 

1422.2 

434 

1607.4 

484 

1792.6 

285 

1055.6 

335 

1240.7 

385 

1425.9 

435 

1611.1 

485 

1796  3 

286 

1059.3 

336 

1244.4 

386 

1429.6 

436 

1614.8 

486 

1800.0 

287 

1063.0 

337 

1248.2 

387 

1433.3 

437 

1618.5 

487 

1803.7 

288 

1066.7 

338 

1251.9 

388 

1437.0 

438 

1622.2 

488 

1807.4 

289 

1070.4 

339 

1255.6 

389 

1440.7 

439 

1625.9 

»89 

1811.1 

290 

1074.1 

340 

1259.3 

390 

1444.4 

440 

1629.6 

490 

1814.8 

291 

1077.8 

341 

1263.0 

391 

1448.2 

441 

1633.3 

491 

1818.5 

292 

1081.5 

342 

1266.7 

392 

1451.9 

442 

1637.0 

492 

1822.2 

293 

1085.2  : 

343 

1270.4 

393 

1455.6 

443 

1640.7 

493 

1825.9 

294 

1088.9 

344 

1274.1 

394 

1459.3 

444 

1644.4 

494 

1829.6 

295      1092.6 

345 

1277.8 

395 

1463.0 

445 

1648.2 

495 

1833.3 

296      1096.3 

346 

1281.5 

396 

1466.7 

446 

1651.9 

496 

1837.0 

297      1100.0 

347 

1285.2 

397 

1470.4 

447 

1655.6 

497 

1840.7 

298      1103.7 

348 

1288.9 

398 

1474.1 

448 

1659.3 

498 

1844.4 

299 

1107.4 

349 

1292.6 

399 

1477.8 

449 

1663.0 

499 

1848.2 

300 

1111.1 

350 

1296.3 

400 

1481.5 

450 

1G66.7 

500 

1851.9 

260  TABLE  13.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 

H: 

Cubic 
Yards. 

Area. 

1?: 

Cubic 
Yards. 

Area. 

& 

Cubic 
Yards. 

Area. 

1?: 

Cubic 
Yards. 

Area. 

8t 

Cubic 
Yards. 

501 

1855.6 

551 

2040.7 

601 

2225.9 

651 

2411.1 

701 

2596.3 

502 

1859.3 

552 

2044.4 

602 

2229.6 

652 

2414.8  I 

702 

2000.0 

503 

1863.0 

553 

2048.2 

603 

2233.3 

653 

2418.5  i 

703 

2603.7 

504 

1866.7 

554 

2051.9 

604 

2237.0 

654 

2422.2 

704 

2607.4 

505 

1870.4 

555 

2055.6 

605 

2240.7 

655 

2425.9 

705 

2611.1 

506 

1874.1 

556 

2059.3 

606 

2244.4 

656 

2429.6 

706 

2614.8 

507 

1877.8 

557 

2063.0 

607 

2248.2 

657 

2433.3 

707 

2618.5 

508 

1881.5 

558 

2066.7 

608 

2251.9 

658 

2437.0 

708 

2622.2 

509 

1885.2 

559 

2070.4 

609 

2255.6 

659 

2440.7 

709 

2625.9 

510 

1888.9 

560 

2074.1 

610 

2259.3 

660 

2444  .4 

710 

2629.6 

511 

1892.6 

561 

2077.8 

611 

2263.0 

661 

2448.2 

711 

2633.3 

512 

1896.3 

562 

2081.5 

612 

2266.7 

662 

2451.9 

712 

2637.0 

513 

1900.0 

563 

20&>.2 

613 

2270.4 

663 

2455.6 

713 

2640.7 

514 

1903.7 

564 

2088.9 

614 

2274.1 

664 

2459.3 

714 

2644.4 

515 

1907.4 

565 

2092.6 

615 

2277.8 

665 

2463.0 

715 

2648.2 

516 

1911.1 

566 

2096.3 

616 

2281.5 

666 

2466.7 

716 

2651.9 

517 

1914.8 

567 

2100.0 

617 

2285.2 

667 

2470.4 

717 

2655.6 

518 

1918.5 

568 

2103.7 

618 

2288.9 

668 

2474.1 

718 

2659.3 

519 

1922.2 

569 

2107.4 

619 

2292.6 

669 

2477.8 

719 

2663.0 

520 

1925.9 

570 

2111.1 

620 

2296.3 

670 

2481.5 

720 

2666.7 

521 

1929  6 

571 

2114.8 

621 

2300.0 

671 

2485.2 

721 

2670.4 

522 

1933.3 

572 

2118.5 

622 

2303.7 

672 

2488.9 

722 

2674.1 

523 

1937.0 

573 

2122.2 

623. 

2307.4 

673 

2492  6 

723 

2677.8 

524 

1940.7 

574 

2125.9 

624 

2311.1 

674 

2496.3 

724 

2681.5 

525 

1944.4 

575 

2129.6 

625 

2314.8 

675 

2500.0 

725 

2685.2 

526 

1948.2 

576 

2133.3 

626 

2318.5 

676 

2503.7 

726 

2688.9 

527 

1951.9 

577 

2137.0 

627 

2322.2 

677 

2507.4 

727 

2692.6 

528 

1955.6 

578 

2140.7 

628 

2325.9 

678 

2511.1 

728 

2696.3 

529 

1959.3 

579 

2144.4 

629 

2329.6 

679 

2514.8 

729 

2700.0 

530 

1963.0 

580 

2148  2 

630 

2333.3 

680 

2518.5 

730 

2703.7 

531 

1966.7 

581 

2151.9 

631 

2337.0 

681 

2522.2 

731 

2707.4 

532 

1970.4 

582 

2155.6 

632 

2340.7 

682 

2525.9 

732 

2711.1 

533 

1974.1 

583 

2159.3 

633 

2344.4 

683 

2529.6 

733 

2714.8 

534 

1977.8 

584 

2163.0 

634 

2348.2 

684 

2533.3 

734 

2718.5 

535 

1981.5 

585 

2166.7 

635 

2351.9 

685 

2537.0 

735 

2722.2 

536 

1985.2 

586 

2170.4 

636 

2355.6 

686 

2540  7 

736 

2725.9 

537 

1988.9 

587 

2174.1 

637 

2359.3 

687 

2544.4 

737 

2729.6 

538 

1992.6 

588 

2177.8 

638 

2363  0 

688 

2548.2 

738 

2733.3 

539 

1996.3 

589 

2181.5 

639 

2366.7 

689 

2551.9 

739 

2737.0 

540 

2000.0 

590 

2185.2 

640 

2370.4 

690 

2555.6 

740 

2740.7 

541 

2003.7 

591 

2188.9 

641 

2374.1 

691 

2559.3 

741 

2744.4 

542 

2007.4 

592 

2192.6 

642 

2377.8 

692 

2563.0 

742 

2748.2 

543 

2011.1 

593 

2196.3 

643 

2381.5 

693 

2566.7 

743 

2751.9 

544 

2014.8 

594 

2200.0 

644 

2385.2 

694 

2570.4 

744 

2755.6 

545 

2018.5 

595 

2203.7 

645 

2388.9 

695 

2574.1 

745 

2759.3 

546 

2022.2 

596 

2207.4 

646 

2:592.6 

696 

2577.8 

746 

2763.0 

547 

2025.9 

597 

2211.1   ; 

647 

2396.3 

697 

2581.5 

747 

2766.7 

548 

2029.6 

598 

2214.8 

648 

2400.0 

698 

2585.2 

748 

2770.4 

549 

2033.3 

599 

2218.5 

649 

2403.7 

699 

2588.9 

749 

2774.1 

550 

2037.0 

600 

2222.2 

650 

2407.4 

700 

2592.6 

750 

2777.8 

TABLE  13.— CUBIC  YARDS  IN  100  FEET  LENGTH.  261 


Area. 

!?: 

Cubic 
Yards. 

Area. 

§q- 
tt. 

Cubic 

Yards. 

Area. 
SSq. 

i   F<- 

Cubic 

Yards. 

Area. 

ft 

Cubic 
Yards. 

Area. 

St 

Cubic 
Yards. 

751 

2781.5 

801 

2966.7 

851 

3151.9 

901 

3337.0 

951 

3522.2 

752 

2785.2 

802 

2970.4 

852 

3155.6 

902 

3340.7 

952 

3525.9 

753 

2788.9 

803 

2974  .  1 

8o3 

3159.3 

9U3 

3344.4 

953 

3529.6 

754 

2792.6 

8o4 

2977.8 

854 

3163.0 

904 

3348.2 

954 

3533.3 

755 

2796.3 

805 

2981.5 

8f>5 

3166.7 

905 

3351.9 

955 

3537.0 

756 

2800.0 

806 

2985.2 

856 

3170.4 

906 

3355.6 

956 

3540.7 

757 

2803.7 

807 

2988.9 

857 

3174.1 

907 

3359.3 

957 

3544.4 

758 

2807.4 

808 

2992.6 

858 

3177.8 

908 

3363.0 

958 

3548.2 

759 

2811.1 

809 

2996.3 

859 

3181.5 

909 

3366.7 

959 

3551.9 

760 

2814.8 

810 

3000.0 

860 

3185.2 

910 

3370.4 

960 

3555.6 

761 

2818.5 

811 

3003  7 

861 

3188.9 

911 

3374.1 

961 

3559.3 

762 

2822  2 

812 

3007.4 

862 

3192.6 

912 

3377.8 

962 

3563.0 

763 

2825.9 

813 

3011.1 

863 

3196.3 

913 

3381.5 

963 

3566.7 

764 

2829.6 

814 

3014.8 

864 

3200.0 

914 

3385.2 

964 

3570.4 

765 

2833  3 

815 

3018.5 

865 

3203.7 

915 

3388.9 

965 

3574.1 

766 

2837.0 

816 

3022.2 

866 

3207.4 

916 

3392.6 

966 

3577.8 

767 

2840.7 

817 

3025.9 

867 

3211.1 

917 

3396.3 

967 

3581.5 

768 

2844  4 

818 

3029.6 

868 

3214.8 

918 

3400.0 

968 

3585.2 

769 

2848.2 

819 

3033.3 

869 

3218.5 

919 

3403.7 

969 

3588.9 

770 

2851.9 

820 

3037.0 

870 

3222.2 

920 

3407.4 

970 

3592.6 

771 

2855.6 

821 

3040.7 

871 

3225.9 

921 

3411.1 

971 

3596.3 

772 

2859.3 

822 

3044.4 

872 

3229.6 

922 

3414.8 

972 

3600.0 

773 

2863.0 

823 

3048.2 

873 

3233.3 

923 

3418.5 

973 

3603.7 

774 

2866.7 

824 

3051.9 

874 

3237.0 

924 

3422.2 

974 

3607.4 

775 

2870.4 

825 

3055.6 

875 

3240.7 

925 

3425.9 

975 

3611.1 

776 

2874.1 

826 

3059.3 

876 

3244.4 

926 

3429.6 

976 

3614.8 

777 

2877.8 

827" 

3063.0 

877 

3248.2 

927 

3433.3 

977 

3618.5 

778 

2S81.5 

828 

3066.7 

878 

3251.9 

928 

3437.0 

978 

3622.2 

779 

2885.2 

829 

3070.4 

879 

3255.6 

929 

3440.7 

979 

3625.9 

780 

2888.9 

830 

3074.1 

880 

3259.3 

930 

3444.4 

980 

3629.6 

781 

2892.6 

831 

3077.8 

881 

3263.0 

931 

3448.2 

981 

3633.3 

782 

2896.3 

832 

3081.5 

882 

3266.7 

932 

3451.9 

9«2 

3637.0 

783 

2900.0 

833 

3085.2 

883 

3270.4 

933 

3455.6 

983 

3640.7 

784 

2903  7 

834 

3088.9 

884 

3274.1 

934 

3459.3 

984 

3644.4 

785 

2907.4 

835 

3092.6 

885 

3277.8 

935 

3463.0 

985 

3648.2 

786 

'^91  1  .  1 

836 

3096.3 

886 

3281.5 

936 

3466.7 

9*6 

3651.9 

787 

2914.8 

837 

3100.0 

887 

3285.2 

937 

3470.4 

987 

3655.6 

788 

2918.5 

838 

3103.7 

888 

3288.9 

938 

3474.1 

988 

3659.3 

789 

2922.2 

839 

3107.4 

889 

3292.6 

939 

3477.8 

Q?9 

3663.0 

790 

2925.9 

840 

3111.1 

890 

3296.3 

940 

3481.5 

990 

3666.7 

791 

2929.6 

841 

3114.8 

891 

3300.0 

941 

8485.2 

991 

3670.4 

792 

2933.3 

842 

3118.5 

892 

3303.7 

942 

3488.9 

992 

3674.1 

793 

2937.0 

843 

3122.2 

893 

3307.4 

943 

3492.6 

993 

3677.8 

?94 

2940.7 

844 

3125.9 

894 

3311.1 

9)4 

3496.3 

994 

3681.5 

795 

2944.4 

845 

3129.6 

895 

3314.8 

945 

3MX).0 

995 

8685.2 

796 

2948.2 

846 

3133.3 

896 

3318.5 

946 

3503.7 

996 

3688.9 

797 

2051.9 

847 

3137.0 

897 

3322.2 

947 

3507.4 

997 

3692.6 

798 

2955.6 

848 

3140.7 

•     808 

3325.9 

948 

3511.1 

998 

3696.3 

799 

2959.3 

849 

3144.4 

899 

3329.6 

949 

3514.8 

999 

3700.0 

800 

2963.0 

850 

3148.2 

900 

3333.3 

950 

3518.5 

1000 

3703.7 

262 


TRIGONOMETRIC   FORMULAS. 


TABLE  14. — TRIGONOMETRIC  FORMULAS. 


FIG.  98. 


RIGHT    TRIANGLES. 
Z> 


FIG.  99, 


In  Fig  99,  let  DCE  be  the  arc  of  a  quadrant,  ABC  a  right 
triangle,  the  angle  13 A  C  subtended  by  the  arc  CE  =  A,  and 
consider  the  radius  A  C  =  unity.  Then 


BC  =  am 

AB  =  cosA. 
HE  =  tan  A. 
DF  =cotA. 
AH  =  sec  A. 


^ljP=cosec^.. 
J5£r  =  versinJ.. 
DI  =  co  versing, 
CH=  exsec-4. 
CF  =  coexsec^.. 


Using  the  small  letters  a,  &,  c,  to  represent  the  sides  of  a 
right  triangle  in  Fig.  98  or  99,  we  may  write 

sin^l  =  -;  eosec.4— -;  .-.  sin>4  = ... 

6  a  cosec^l 

=  -;      sec  A—--,   .'.cosA= 

6  c  sec  A 

=  -:      cot  A  =-;.'.  tan  A  =  — L— . 
c  a  cot-4 


OBLIQUE  TRIANGLES  263 

FORMULAS   FOR   LOGARITHMIC   COMPUTATION. 

Given  one  side,  a,  and  the  three  angles  A,  B,  and  C. 

The  "sine  proportion"  states  that  each  side  of  a  triangle  is 
proportional  to  the  sine  of  the  angle  opposite  it. 

™,  ,       a  sin  B         .    „       a  sin  C 

Then,        b  =  —-. — -r-     and    C  = 


Area 


.  cauu      \s      -  . 4    . 

sm  A  mnA 

a2  sin  B  sin  C 


2  sin  (B+  C) 
Given  two  sides  a  and  6  and  the  included  angle  C. 

Ftrs/  method: — Determining  the  angles  first. 

$(A  +  B)  =  J  (180  -  C) 
tan  \(A  -  B)  =  ~-^  tan  J  (A  +  B). 

A  =  J  (A  +  B)  +  |  (A  -  B) 
B  =  i  (A  +  B)  -  $(A  -  B). 

The  third  side,  c,  can  now  be  determined  by  the  sine  proportion. 
Second  method: — Determining  the  third  side  first. 

2  sin  i  C     /-- 
tan  x  =  —  -  Va  h. 

a  —  b 

a  -  b 
c  =  — 

COS  X 

The  angles  A  and  B  can  now  be  found  by  the  sine  proportion 

Area  =  J  a  b  sin  C. 
Given  the  three  sides  a,  6,  and  c. 

Let  s  =  >-  (a  +  6  -f  c). 


\|  —  (s  —  a)  (s  —  b)  (s  —  c) 
Then      *tan  \A  =  — — 


*tan 


J  -i-  (s  -  a)  («  -  6)  (.s  -  c) 


-  (s  —  a)  (s  —  b)  (s  —  c) 


Area  =  -y/  s  (s  —  a)  (s  —  6)  (s  —  c) 


*  It  will  be  noted  that  the  numerator  is  the  same  in  each  case  and 
that  the  denominators  are  quantities  whose  logarithms  are  known  and 
hence  two  additional  logarithmic  operations  over  that  required  for  one 
angle  gives  all  three  angles  and  a  complete  check  on  the  entire  compu- 
tation since  A  +  B  +  c  =  180°. 


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